I've read your post and you do not sound ungrateful or bratty at all.
As a Muslim myself, growing up I did have lots of questions asked when I'd be going out but that was out of curiosity and making sure I was safe, because my parents are really overprotective of me - being Muslim has nothing to do with it at all. I feel as though this is the case for you because there is no real basis for which they are keeping you inside and whatever reasoning they would have would just be silly.
You're going to have to do these things whether they like it or not, don't even ask them anymore just tell them you're going out as soon as you get ready so they have less time to interfere; because this is obviously harming you mentally and emotionally. You could try and tell them the positives of going out and give them reasons such as; health, exercise, studying etc. Really sell the idea to them that all these things are for you future and that they're holding you back tell them of all the thing s you could have done but haven't been able to do because of them. Don't let yourself suffer, or your education, or your career, or your mental state just so that your parents don't get angry - does that really seem worth it to you?
But in regards to the university thing you have to socialise especially for meetings with lecturers and groupwork along with trips to the library for your own learning - it's not really an option to do it from home and if you really wanted to do that I would suggest doing a course with the Open University. Try questioning your parents at what they want you to achieve? Staying at home isn't going to do much for you and you can't spend the rest of your life like this especially because you're an adult. You need to sit them down and talk to them, but there isn't room for negotiation in this; it's them having to understand that you need to do these things. I guess you could ask your older sibling to go with you so that your parents can ease into this change, but that is your choice. If they compare you to your cousins just tell them if they let you go out you can do even better!
There is the idea of moving out and that is completely fine to do, but beneath all this I feel that deep down you don't want to resort to that and that you do care quite a bit for your parents. It's a good thing that you're already able to go out for your job, but you really need to stay stubborn in wanting to go out more often. Your parents are so bothered about the idea of a better future but what future is there by staying inside surrounded by 4 walls. Save up for your future, take online courses to build up your cv, do therapy online if that helps and plan a future for yourself. You either ignore them and do as you please, move out or just accept it - I wish you the best of luck and hope you feel better soon.