Disclaimer: Med student making connections that are by no means guaranteed to be 100% correct. Sorry this turned out a lot longer than I intended.
I'd say go for it! The gym will help with your depression for one thing. The increase in the hGH you'll produce will be beneficial as it has a natural mood elevating qualities (if I recall correctly), granted you won't be shooting your levels through the roof. The improved physique will also help with your mood (humans are vain creatures after all). This alone should be reason enough to take advantage of the gym.
Squat and deadlifts are great for the behind. I don't know the extent of your muscle involvement from your childhood infection, but said you had difficulties with the affected side. Start slow and work into it to gauge the extent of your muscular involvement. It may be nothing, or maybe your muscles were involved during your infection. The difference between the two sides can be due to a number of little reasons adding together making one side weaker and/or less flexible than the other. If you don't mind me asking, do you remember what infection it was?
Assuming all is well with your gluts, and after a bit of time, the muscle mass will help with the shape of your bottom. Stick with it and the muscle help to tighten the skin, possibly reducing the noticeability of the scar. It is hard to say just how dense the scar tissue is from the pictures. The more dense, the more work it may take, but you WILL see a difference with time. Understand though, the scar itself won't go away, it would just be "stretched out" over the muscle.
As you might be able to tell by my wording, I'm trying not to make false promises by giving definitive statements. Still, simply put, do what you can, even if you had some muscular involvement, it's not likely to be much since you haven't mentioned (as far as I know) any disabilities. Like your GP said, the butt is a big muscle group. Set some realistic goals in the beginning (run for this long or do this many squats) and keep track of it. It makes it easier to progress if you write down the numbers. It's also easier to see your progress. It helped me stay consistent with my schedule because I saw I was achieving my goals. So just ease into, eat right (protein for muscle growth, veggies for health), and let yourself be happy! Kick depression's ass and while you grow a new one!
TL;DR - Start slow to gauge the effects your infection had. Go for it regardless of your bum, it will help your depression. If you have any other questions, send me a message and I'll help the best I can.