What R1 said.
Regarding private flying, if you want to spend the money at the flying club, sure, knock yourself out. Once you're actually in the program, you are unlikely to have much free time on your hands, but if you're stashed for a while, no reason you can't.
As for your Xtreme Sports, I was the NASC Safety-O during my instructor tour. Say you're skydiving and shatter your knee, so you're injured 'not in the line of duty', and it's bad enough to put you on LIMPDU indefinitely. Now you're in the fantastic world of Med Hold, which operates outside normal space-time. You're out of training and manning a desk or fetching coffee for somebody, and that's
after you're healed up. Then you're waiting for some random council of quacks to pronounce you ready to rejoin the program. This process can and has taken a year or more to play out. Sometimes a lot more.
At some point the Skipper/CDRE/CNATRA is going to start considering whether it's worth keeping you in the program. That's not punishment; there are only so many flight school billets to go around, and they have a production target to meet. They have to assess whether it's worth keeping a spot that could be used by someone healthy, when they haven't spent much money on you yet in training and you may not ever be a full-up round again. That you got hurt in a recreational off-duty mishap definitely won't help. And if you're out of the program and nobody else needs an unqualified gimpy ensign, then the Navy is going to start considering whether it's worth keeping you in at all, or just sending you home. I have seen this happen.
Caveat, because apparently a lot of studs misunderstood my speech about this: you will
not automatically be attrited for getting sick, or for minor injuries. If you get hurt or ill, fess up and go see the Doc. "Toughing it out" is dangerous and fucking stupid, and will not impress anyone. I'm talking about serious injuries, occurring during off-duty shenanigans and/or monkeyshines.
Okay, climbing down now.