Colleges in West Bengal - Other

Yes, you are eligible for Asansol government college.

Asansol engineering college is better for you.

Yes, college allow hostel students to move out for thier need.

Initially, you might not get hostel allotment as already regular students get alloted and stays in the second year hostel.
But there's chance of some vacancies so you might contact the proctoral section for details. If there's vacancy then you might get a chance to stay in hostel.
Ekse you have to manage in nearby PGs or mess. That is also affordable.

I think you might enter with GMR (although slight chances). You'll get IT, with your GMR I think. (spot counselling might or might not happen this year, where you can upgrade to ECE but there are chances of some online counselling for vacant seats, Everything is so uncertain this year) Please look into the detailed seat matrix for this year before filling up. (you will find it any of the WBJEE sites)
Always double check your category and seats present in a particular stream for that category. It changes sometimes every year.