I don't see any reason you wouldn't be able to dive. Since you don't have any medical conditions, you should be fine there. It sounds like some instructors do require a doctor's clearance even if you don't mark anything on the medical form, but certainly plenty don't (mine didn't), and considering that you're a healthy 15-year-old, I don't see any reason you'd need to be checked out by a doctor before you can dive unless your instructor requires it. And really there's not much that your doctor could find that he wouldn't have found at a regular check-up or that would be a concern for diving but not even worth mentioning otherwise (especially since things like even mildly high blood pressure that could be risk factors for diving and are relatively normal for, say, middle-aged adults would still be unusual enough in a teenager to have discussed at a normal check-up and not just in the context of diving). It's more just if you had any chronic medical conditions, or for older people who are at higher risk for heart problems and strokes and stuff. For the most part though, and medical problems that would be an issue for diving would also be an issue for other sports, it's just that something that might make you pass out or even have a seizure while you're playing basketball or whatever, you might end up with a few bruises when you hit the ground, but the same thing happens 60 feet underwater and there's a good chance you'll drown.
As for buying equipment, most I'll just echo what others have said about waiting until you're at least certified (in fact, many places wouldn't even sell you most gear until then). But also, when you are ready to think about buying gear, keep in mind that at 15, you're probably still growing, and even if you don't end up getting much taller, you may still gain or lose weight and just sort of generally change shape a bit over the next few years. Some things like a reg and dive computer that won't matter, but for thing wetsuits especially and to some extent BCs as well, it's worth considering. When I got my BC, I (or rather my parents) ended up having to spend quite a bit for one that worked well for me (fit and various other things -- there's a lot more than just finding the right size), but for something I'll probably be using for at least 10-20 years it's not bad at all, whereas if I might only get to use it for a year or two before it didn't fit quite right it would have been way too much to have spent on it (even if it was still usable, if it just wasn't the best style for me anymore, I would have been a lot better off if I'd waited longer to invest in one and gotten a decade or two of really good wear out of it instead of a year or two of good wear and several more of acceptable wear). So when you're thinking about buy gear, assuming that you can't afford to replace it in another year or two (and probably once more after that), you might want to consider the possibility of renting some of it for the first few years, and waiting to buy things until you can be sure you won't grow out of it.