I don't jump much because my scoliosis has resulted in fusion of some of my lower vertebrae, so I now lack the 'bounce' that a properly curvy spine would give me. But I still jump. I just don't jump high. And I try not to jump too much at one go, especially on a non-rebounding surface.
Superman stretches for me are a real struggle, but they won't damage me while they strengthen the muscles in my lower back, so I work them. I hate them, though.
Exercises that strengthen the back are a good idea so that you hold your spine in a good position. Mind your posture, especially - well, I was going to say 'when you're sitting' but really, you need to be mindful of how you move all the time. Be aware of when you might be slumping in the middle. Core exercises are good for that.
There are loads of mobility workouts here - you might start with the Mobility Collection - and that's really important.
Not lifting too much weight is probably again related to the lumbar region, but it probably doesn't mean that you can't do things like pull-ups or chin-ups which not only help build that upperbody, but also stretch the spine and strengthen the core.
PT is good, but if you can, perhaps see a rehab specialist, or an exercise specialist. Don't limit yourself by something that you're told at the age of 22. Unless you've been particularly prohibited from doing something and have been given a really good reason why, move as you feel able and try to move outside your comfort zone so long as you don't cause yourself sharp or ongoing pain. A good chiropractor can also be useful to have in your contacts list. You've got a lot of years ahead of you and you've got to keep that back - and all the other bits around it like core and hips and shoulders - strong.