I was on depo provera for a year or so. Periods stopped immediately, and didn't come back until I'd been off it a few months.
I went off of it because I didn't like the side effects I got from it. When I got a shot, I'd be hungry and really moody for a few days, and after that I still didn't exactly feel like myself. I did gain weight, but in my case that's a good thing. I can't remember if there were any other problems, but I remember being unhappy with it enough that I was willing to keep having periods rather than deal with it.
I wouldn't approach an obgyn with any requests specifically to stop bleeding. That tends to get dismissed or treated with suspicion. Just tell the doctor you need more effective birth control, and that you've heard good things about Depo Provera.
Also, I had no feminizing effects from it. I think it's worth trying. I very much loved not having periods, and not everyone has trouble with the drug like I did.