Ask your coach what you need. They may have rules in regards to gis, etc.
But, FWIW, cups are “illegal” in BJJ tournaments.
I don’t know how many guys wear them in training, but they are definitely a minority because in grappling a cup is probably just as likely to cause you injury as to prevent it. A hard one can also mess up your partner. Like any sport, things can happen but I’ve never felt the need to wear one. Wore one a few times in the beginning and ditched it.
I tried a regular cup and didn’t like it so I don’t wear it anymore. As for the mouthgaurd, I’d say to invest in one - esp since the custom fit ones feel significantly better. I think it’s FSA eligible now, too.