I shave about once a week. Like others have said, it helps prevent odors - before I started shaving, I would fairly regularly get a whiff of urine smell after I'd cleaned up from an accident, but that's much less of a problem now.
I use a razor in the shower, which seems to be the easiest for me. That eliminates any problems with leaks while shaving. I put a bit of soap on the area and shave while I face away from the shower head, then when I'm done I can just turn around and wash off any soap residue.
I also use a little bit of underarm deodorant in areas where my skin folds to help ward off the "sweaty body" smell that I can get when it's hot. It's lightly scented but not overpowering, and seems to help a lot. I don't know if anybody else does this - it feels a little strange to be putting deodorant on "down there," but it sure helps me!
I used to be shy about doctor visits after shaving, but I've gotten over that. I've never had a comment on it from a doctor, and I suspect they see lots of people who shave anyway.