As far as I know, 15-18 schools sounds pretty standard. I originally applied to 22, cut it to 15. I now wish I'd only applied to about 8 or 10. It costs a lot to apply and if you know where you definitely want to go, there's no reason to apply elsewhere if you are a decent candidate. Having visited schools, I know I definitely am more interested in osteopathic schools and I know I want to get out of NYC. Knowing that beforehand would have made me much more efficient by applying to different schools (and less of them) than I already have.
As far as schools knowing where else you applied, that's a good question. There's probably some way they can find out if they really want to, but, in my experience, they just ask you where you have applied in the interview anyway. I think it's pretty terrible when they do that, especially for me because I applied to more schools that I'm not interested in than those that I am. This always raises more questions and interviews always become a big mess. I don't think they hold it against me, though.
My point: if you have 15-18 schools you think you like, research them like hell, talk to people, but most importantly, visit if you can. I think that med schools are interested in where else you applied and why, but overkill is common and I don't think they'll hold it against you. 15-18 is not overkill.
-RDO