I went to art school for 5 years, I took MANY figure drawing classes & loved every one. The very first nude figure drawing assignment we had was awkward, it just is, it can't be helped, but we quickly got comfortable & learned to appreciate doing it, it was a definite favorite class of mine. It IS a touchy subject just because MOST people don't understand the concept of nudes in art. But it doesn't sound like you're going to be modeling for first time figure artists, so you will most likely feel comfortable with them.
I've had all sorts of models, from perfect athletic built 18 year olds, to old saggy great grandmas (ewe, yes, not my favorite) it's not that artists are looking for the perfect body, we're looking for aesthetics of a figure & since people come in all shapes & sizes it was nice having a variety. I tend to prefer the average (not skinny) female figure in their 30-40's. But that's just preference. So don't ever worry that you'll be judged about your physique.
I can't really comment on the "other matter". I can't count how many nude models I've used, most of them were women, but if I had to guess I'd say 20 different men & only ONCE did that "other matter" happen. To be honest it made me very uncomfortable & I even left the room but this was because the guy was smirking about it & it was creepy. If you are worried about this happening, I would talk to the art professor who approached you & discuss it seriously with him. I'm SURE he will be able to advise you on this matter & even tell you what other male models do to avoid this. (I hope you're serious & not being funny.)
Usually, these classes are done with extreme privacy (closed doors & windows) & is very respectfully done. I know artists appreciate & respect the models who do this, they DO make good money for this & I think they deserve every penny. So if you're comfortable with it all, go for it! I don't think you'll regret it.