Hey there,

yes a student can surely can give jee mains and advanced exam while pursuing study in some other colleges. Generally, the colleges take tuition fee yearly or semester wise.  Thus, you do not have to pay your whole 4 year fees to get your document back instead you should pay the fees only till the time you study in that institution.

If you get a good rank in the jee mains examination then certainly you can take admission in some other colleges by taking part in the counselling.

Yes the college give back the documents required to take admission in the other college. You generally have to take it from your head of department and get it signed by the dean of academics.

Hope this resolves your query.