So I know that I implied that this was just going to be a one-shot, but I managed to get some ideas for it that just would not leave me alone. So here you go. Hopefully this helps answer a few of the questions you had during the first part.

A few days after the party, Severus had finally gotten the house back to the state it was in before the party. He had just sat down to look through a new potions book he had bought a couple weeks prior when someone knocked at the door. He put the book back on the table and went to answer the door.

"Hey, Lily," he said and stood aside so she could come in. "I just got Sabrina down for her nap."
"Alright. Actually, I came by to tell you something," she said. "Or rather, I need advice." He looked curious and motioned for her to tell him. She took a deep breath and said, "I'm pregnant. I found out a couple days before Sabrina's birthday, but I wasn't ready to tell anyone. Not even James knows yet."
"Congratulations," Severus said and hugged her.
"Thanks," she replied then pulled back a little, but still held on to him. "I don't know how to tell James. How did you tell Sabrina's dad about her? You… did tell him, right?" Severus laughed lightly.
"Yeah, I told him. Alright, have a seat. I guess its story time, but be warned, Regulas wasn't happy about it."

Flashback

For a few weeks now, Severus had been feeling bad, usually only in the mornings, but sometimes it lasted all day. He was about to consider going to Madam Pomfrey, but was going to hold off a little longer since he was sure it was just a really long bout of food poisoning or something. However, after a few days, he couldn't wait any longer.
Once he arrived there, Madam Pomfrey asked him a few questions then handed him a container. "What's this for?" He asked her. She looked amused.
"I think I know what's wrong with you, but I need a urine sample. And since you just told me, you need to urinate more often, it shouldn't be to hard." He sighed, but walked into the adjoining bathroom. A few moments later he walked back out with the cup and placed it in front of her. "Thank you." She waved her wand over it and frowned. She waved her wand again to the same result. She looked over at him and back at the container then waved her wand over him. "Huh. Well, congratulations, Mr. Snape. Your pregnant." He jumped up.
"That's impossible. I'm a boy, boys don't get pregnant," he all but shouted. She looked puzzled.
"Were you raised in a Muggle household?" Severus stared at her.
"What does that have to do with anything?"
"Just answer the question, Mr. Snape." He nodded. "I thought as much. You know that the muggle and wizarding world are different in a lot of ways. Well, one of those ways is that wizards can get pregnant. Well, certain wizards can anyways. The general theory is that a wizard many centuries ago created a potion to induce male pregnancy by creating a uterus. And after so many generations used it, it eventually turned into a mutation. Now, certain Pureblood lines have this ability. Are either of your parents pureblood?"
"Yeah, my mom was a Prince." Madam Pomfrey nodded.
"I believe that is one of the families. Now, you'll be needing a Prenatal Potion, which work just like a Muggle Prenatal Vitamin. And," she wrote something on a piece of paper, "give this to Professor Slughorn. He can make you a potion that will help with the morning sickness. Do you have any questions?"
"Um… can you tell how far along I am?" He asked.
"Sort of. You're in the beginning stages so its hard to tell exactly at this point, but I'd say about six or seven weeks. I'll be able to tell exactly at your first scan. I'd do one now, but I don't have the proper equipment. I'll most likely have a Healer in Hogsmead do the scans. I'll owl you when I know for certain"
"Alright," Severus replied. After a reminder to be careful about what to do and eat, Severus left the hospital wing feeling worse than he did going in. How was he supposed to tell Regulas about this? They had agreed in the beginning that this was just going to be casual sex. No relationship, no feelings, and certainly no children. How could he screw things up so completely? He sighed. Well, first things first, he needed to take this note to Slughorn. He could already feel another bout of sickness coming on.

After getting the potion from Slughorn, Severus felt quite a bit better. He was even starting to get use to the idea of having a child of his own. He wasn't entirely confident in his abilities, but he was certain that he would be more attentive and loving than his own parents had been. Now, he just had to tell Regulas about the child. But how to go about that?

A few days later he got his answer. Ever since they created this little arrangement, whenever they wanted to get together, they sent an owl with the time and place. Today, Severus got one such owl. All it had said was abandoned fourth floor corridor, midnight, but it offered Severus the perfect excuse. And maybe, he would wait until after the deed for which he was called to tell him. Regulas was always more pleasant after sex.
Later that night, Severus left the Slytherin Common Room close to eleven thirty with the excuse that he was going to the kitchens for a late night snack. As he was walking toward the meeting place, he started thinking about what Regulas might say. He highly doubted that Regulas would be accepting. Regulas was a good person, but he was also too attached to his mother to do anything she would disapprove of. Like take responsibility for a Half-Blood child. He shook his head, at times he both loved and hated that boy, but regardless. Maybe Regulas would send money for child support? He almost laughed. Unless Wallaburga specifically said he could, Severus highly doubted he would see a knut of the Black Family money. Unless he threatened to reveal the child's blood status? They'd probably pay him just to shut up about it. Now there was an idea. Hm.

Whenever he finally got to the meeting spot, Regulas was already there, looking highly impatient. "Where have you been? It's almost twelve fifteen." Ah one of the things he hated.
"I saw Mrs. Norris and had to take the long way around." A pretty good excuse considering it was the first thing that popped into his head. Regulas just nodded and motioned for Severus to follow him.

About an hour later, the two were curled up in a makeshift bed together. Severus looked up at Regulas and opened his mouth to tell him about the baby, but decided against it when Regulas looked at him. Even though it went against the agreement, somewhere along the line, Severus had grown to care for the younger man. He wouldn't call it love, but maybe somewhere close. After raising up and giving the other male a light kiss, he lay back down and fell into a restless sleep, deciding to wait another day to tell him about the baby.

When Severus woke up, he realized a few things. 1.) They hadn't gone back to the dorms last night so Severus's dorm mates were going to kill him, 2.) he forgot to bring the potion with him and was starting to feel the effects of it, and 3.) he didn't tell Regulas last night. But due to the morning sickness, the other two things took a backburner. Severus ran over to one of the sinks in the classroom they had found last night and proceed to lose what little he had managed to eat at dinner last night. Regulas cringed, but went over to his lover and held his hair back while he retched. A moment later, Severus felt it was safe to raise up. Regulas handed him a wet paper towel which Severus used to wipe his mouth with. "I thought you went to the Hospital Wing about that?"
"I did. It's normal," Severus replied then rinsed his mouth out.
"Puking your guts up it normal? Since when?"
"Yeah. I just caught a bug, it's not contagious anymore though," he added once he saw the look on Regulas's face. "I swear."
"Ok. If you say so. But if I get sick, no sex until I get better," Regulas said. Severus shrugged. Regulas pouted.
"Are you ready to go? I think I can safely leave the room now." Regulas looked skeptic, but nodded.

When they got back to the dorms, Regulas went up to his room to shower and Severus sat in the Common Room berating himself about not taking advantage of the situation. But Regulas was being so caring, he didn't want to ruin that. But he deserves to know plus he couldn't hide it forever. Soon enough, Severus would start gaining weight. Gaining weight. Pregnant people did that, didn't they? Most people were unaware of this little fact, but Severus Snape had spent so much of his life being starved because of his father alcoholism that he eventually started to think that maybe it was his fault that they were poor. After all, if he didn't need to eat, they wouldn't have to spend so much on food and thus more money would be left over after the alcohol was bought. This ideology lead to Severus becoming somewhat of an anorexic, though he wouldn't admit it to anyone. You know, maybe being pregnant wasn't such a good thing anymore. Shit. No. No. He wanted this child. Even if he had to become fat over it. No. Not fat, pregnant. Damn this was gonna be hard.
Severus snapped out of his inner ramblings when Regulas came back downstairs with water from his hair dripping down his neck heading toward his- Severus shook his head. His hormones must be going crazy if he found that sight arousing. "Regulas," Severus spoke up. Regulas looked at him and smiled. He really was a pleasant person.
"Yeah ,Sev." Oh, he called me Sev. He paused. I'm going crazy.
"I um… can we talk? Not here, but… somewhere more… private?" Severus asked. What the hell are you doing?! What happened to waiting?
"Of course. Would you like to go up to the dorms?"
"No, too many people have access to it. How about somewhere on the upper floors? Where nobody goes?" Regulas looked confused (How adorable), but agreed nonetheless.

They got all the way up to the seventh floor, near a tapestry of a man teaching trolls to dance, before Severus finally stopped. He walked back and forth in front of the tapestry trying to decide how best to tell him when Regulas made a noise of shock (which he would later deny if asked). Severus turned around and saw a door. That wasn't there before. Maybe they could go in there and talk. Severus opened the door and peered in before slamming it shut. Inside the room, was a bunch of baby related items. A crib, and giant ball that Severus couldn't figured out the reason behind, some bottles, and a bunch of other early child care related items. Regulas looked surprised by his reaction, but shook it off and tried to open the door, but Severus wouldn't let him. "Come on, Sev. Let's just go inside." Severus whimpered, but consented.
When Regulas opened the door, he stopped in shock. He turned to look at Severus, "What the hell is this?"
"I- It's a baby room," he looked at the ground.
"Why did you bring me to a baby room?" Severus didn't say anything. Regulas grabbed him by the chin and forced him to look up. Severus had tears in his eyes. "What. Is. This?"
"I- I- I'm p-p-pregnant," he spoke softly. He thought for a second that Regulas didn't hear him until he forcefully pushed Severus away from him.
"It's not mine." Severus looked up in shock.
"What?"
"It's not mine. Are you deaf? I can't be the father of some- some- Half-Blood. It goes against everything I am," Regulas replied
"Yes, it is. I haven't slept with anybody else. Just you. And I'm starting to see that that was a mistake," Severus replied. At this point, he desperately wanted to curl up and cry, but he wouldn't give Regulas the satisfaction of seeing that he was hurting by the younger's words.
"What part of I can't be the father of a Half-Blood do you not understand?"
"You know what, fuck you, Black," Severus said before he turned and left the area.

End Flashback

By the end of the tale, Lily looked like she was about to cry. "That's horrible. Did he ever take responsibility?" Severus shook his head.
"No. As a matter of fact, I haven't spoken to him since that day. The only thing I regret about it is that Sabrina won't ever know her father because he was too prejudiced," Severus replied.
"How come you never told me any of this? I mean it was before our fallout," Lily asked.
"After the way Regulas reacted? I was afraid you would react badly. Especially since I was slightly disgusted by the idea until I really got used to the idea. I didn't want to even think that you could hate me because of something I didn't even known was possible," Severus replied. Lily got up and hugged him.
"You know that's not true. I love children and I love you. At the very least, I would have understood what happened a few weeks?" Severus nodded. "A few weeks later a little better. Do you have any idea how much it hurt knowing that your best friend had just called you a Mudblood?" Severus looked at her in shock. Before he realized he was gay, he would have given anything to hear those three little words from her. And she had a point. In the weeks following their falling out, he would have given anything to be able to tell her how he felt when it happened, or better yet, to tell her about the pregnancy, but when she said she wouldn't forgive him he was afraid that if he told her, she would tell everyone else out of spite. Later on, he realized that she would never do something like that, but by then it was too late. He never stopped being grateful that it was her that found him that night when he went into labor.
"I know that now, but back then I was afraid. I wanted so badly to tell you about the pregnancy and how I was feeling and everything, but after you said you wouldn't forgive me, I… I was afraid you would tell the entire school, or at the very least James since you were getting close to him around that time, out of spite because of what I had done," he replied. Both of them were near tears at this point.
"Severus Tobias Snape, I would never do such a horrible thing. How could you think such a thing? No matter how mad I was at you, I would never do something like that," Lily said, tears rolling down her face at the very thought.
"I'm sorry. So very, very sorry. I just- I thought- I," he couldn't continue. Somehow Lily seemed to understand and just held him tighter.
Ironically, Sabrina picked that very moment to come downstairs. "Mommy? What's wrong? Why you crying?" Severus laughed lightly and wiped his tears away before going to pick up his little girl.
"It's ok, Sweetie. Mommy and Aunty Lily were just talking about something that makes us both sad. It's ok though." Sabrina hugged him tightly around the neck. Lily wiped her tears away when Sabrina wasn't looking.
"I," she hiccupped, "I better be going. James is probably wondering where I am. Bye Sev. Bye Sabrina."
"Bye Lil. Say bye Sabrina," Severus said and walked with her to the door.
"Bye-bye Lily."