If anyone is actually interested in this, I'll be genuinely and happily surprised. But if you don't feel like having a biology lecture outside of class, then I suggest skipping out now. You've been warned.
I think I can explain how Shinichi turns into Conan and back. Not the why as in why everything reacts as it does, but you get the gist. Okay, basically a cell has DNA, and that DNA forms chromosomes while in the process of mitosis, right? At the tips of each chromosome are called telomeres. Telomeres are essentially pieces of unused genetic code that help prevent cancer due to chromosome ends sticking together or something similar. They shorten every time a cell divides, and once they grow too short, the cell can no longer divide and dies by a process called apoptosis. With that information, how does this apply to Shinichi?
Well, the drug somehow causes a widespread cell death (apoptosis). By doing this, the cell basically explodes. Considering the cell is mostly water, there is a lot of excess water, too much for the body to handle. Also, that probably explains the intense pain, probably due to both the spontaneous apoptosis and the excess water. The fever is a bit hard to explain, as is the coughing. I don't know very much about fevers and how they work, but from what I've read, chemicals called pyrogens cause the brain to think the body temperature is lower than it actually is and thus caused the body to heat up. They usually react to tissue damage (normally caused by bacteria), so perhaps that's why Shinichi gets the fever. Maybe his brain interprets the whole process as a massive infection. In any case, the fever helps to quickly rid his body of the excess water. The loss of volume actually allows the shrinking process to begin.
Oh, and an enzyme called telomerase is in charge of occasionally lengthening the telomerase in a normal person so the life of a cell is extended. However, in Shinichi's case, the drug must make the telomerase restore the telomeres in his cells to the length they were at the time he was six. Why that particular year, I've no idea. But in the event that the telomerase were to not restore the telomeres, then Shinichi probably would've died shortly after the first transition, if not had serious health problems because his cells wouldn't be able to perform mitosis anymore.
As for the transition back to Shinichi from Conan, he basically has to go back through the aging process at an exponentially faster rate. This means rapid production and death of cells. It's a delicate process because if the dying cells and the newly produced cells weren't in the right ratio, then Shinichi could easily get cancer from malformed cells and organs. The fever is probably caused by the same reasons as the transition to Conan from Shinichi where the brain interprets the situation as an infection. I still have no explanation for the coughing, though. Hm. Am I missing anything? Probably. I'm quite tired at the moment.
One more thing before I sign out, I also can't explain the reason why the antidote only lasts for about a day or so. The change should be permanent both ways, really. But then again, it is a manga. Anything can happen, and thus does not need to be explained. *rolls eyes* That side effect exists purely for the sake of the manga, I'm sure. But if anyone's got an explanation for that one, I'd be glad to hear it, as well as anything I may have missed. Thank you to all those who read this to the end. ^_^ I'm not an expert, but I suddenly got the urge to write this for no reason. Most of my knowledge comes from a number of websites based on faint memories from my biology class. Chemistry is not my thing, but biology is interesting, so that's why I thought I'd take a stab at explaining Shinichi's transition.
Dedicated to Haibara-san and all the science fanatics out there. I wish I could ask her (Haibara) if it's accurate or not. Sources not directly listed because it's a royal pain trying to get the links to show up. However, should the need or curiosity arise, I could try to send them to whoever wants them through PM. As a generalization of the sites: I used How Stuff Works (Science section), The University of Utah's Learn Genetics, and . If you search those websites, the information I used could probably be found.