Do I have any hope to get into any NIT if I have 64524 rank in JEE Mains under general category? Should I register for JoSAA counselling?

Yes, you can participate in the CSAB special rounds even after accepting a JoSAA seat in the NIT+ system, provided you follow the specific rules. 

No, you need to fill fresh choices for CSAB rounds. The choices filled in JoSAA will not be carried to the CSAB counselling.

Candidates can check the JoSAA cutoff on the official website - josaa.nic.in.

Yes, the JEE Main admit card is needed at the time of JoSAA counselling.

Yes. You can. Both counselling have different purposes. JoSAA is for admission to IIT and NIT+ system, while JAC Chandigarh is for admission to  Dr S. Bhatnagar University Institute of Chemical Engineering and Technology; Panjab University, Chandigarh (Formerly Department of Chemical Engineerin