Headcanons for before the pilot:
Cole and Jay bonded over video games because Jay taught Cole under the guise of being helpful, but really he just wanted a punching bag for his combo moves.
They ate a lot of pasta and sandwiches for a long time, mostly because Cole couldn't cook, Jay never felt like cooking a full meal, and Sensei was never there for long. Cole tried at first, but his efforts were quickly shut down by the two other residents.
When Zane arrived, Jay was ready to strangle him by the second day due to his lack of understanding of puns. Cole had to act as a mediator as Jay explained his jokes through clenched teeth.
When it was revealed that Zane not only loved to cook, but was a master at it, Jay could've cried from happiness. All instances of misunderstood humor were forgiven. Cole wondered if Sensei had chosen him as a ninja solely for his culinary skills.
Cole was alone with Sensei for two weeks before Jay arrived, and it took another month for Zane to show up, and the three of them were together for nearly three months.
Jay wouldn't admit it, but Zane's lack of humor actually grew on him after a while, and he purposefully would make confusing puns to confuse him. Knock-knock jokes were common as well.
Zane did all of the shopping, since he was the most organized and focused when given a task. Cole and Jay were the two idiots in the parking lot speeding down the aisle in a shopping cart.
Zane also did the laundry for the three of them, which led to him accumulating a large amount of change, paperclips, receipts, and even a twenty dollar bill at one point. It was Jay's, and Zane made him do all the chores for a week to get it back.
Cole bought blank nametags one day, and proceeded to make customized labels for the residents of the Monastery. Jay was his signature Motor Mouth, Zane was Slenderman (he didn't understand it), Sensei was Dumbledore, and Cole labelled himself Master Chef. Sensei even wore his for a day, which made Cole happier than it should have.
Family was a seldom mentioned topic around the Monastery. Sensei's brother was evil, Cole would only ever grumble about his dad, and for all Zane knew, he might as well not have any family. Jay always felt bad whenever he absentmindedly started talking about his parents, and tried his hardest not to bring it up.
Sensei would sometimes find the three of them on the gaming couch fast asleep after a movie night, often leaning on each other as they slept. He never mentioned it, so as far as they knew, it was a well kept secret that they didn't mind using each other as pillows.
Jay was prone to waking up after particularly bad nightmares, and he would then wake up Zane for some late night comfort hot chocolate and to talk it out. Zane always said he didn't mind because he didn't need much sleep, but the truth was that he would get nightmares of a similar character back at his village, and he wished someone would have done the same for him.