Vellore

You can get any branch in VIT-Chennai campus.
For Vellore also, you can get most of the branches (maybe except CSE).
Yes, you can also get ECE in Vellore.

Yes, you may get admission in either of the campuses but the fee structure would be quite high.

VIT, Vellore is better than Amrita, Bangalore as in VIT there is tie-ups with many companies and also some companies have their research incubators where students go for the workshops regularly.

No doubt that the VIT is the best college it has good incubation center and many research laboratories but the main problem over there is the fees problem that everybody can't afford.

Hello, hope you are doing well. You must go for VIT, Vellore instead of wasting a year in JEE preparation because Vellore is very good college instead of getting colleges with JEE marks. All the best.

No. You can't get admission for M.Tech. You need to pursue BE or B.Tech degree first.

There is an exam called as VITMEE, which you can give irrespective of whether you have appeared for GATE. The forms are out already on vit.ac.in

Obviously, you will get integrated M.Tech degree which means (B.Tech+M.Tech). Hope this helps.

Let me tell you one thing, only core NIT's have good placement all other are also fine, but not as good compared to VIT University, because VIT University offers good placements in CSE, IT, ECE branch and as compared to NIT Durgapur. Chemical branch is not up to the marks, because in the end branch does matter, so it is better to take VIT.