NIT

The minimum score varies with the year, stream and the IIT. Board marks also have weightage. Scoring a rank under 5000 should do it for you.

Placement wise, if you go for new NIT or IIT (new), there is very less scope. I think you take a 1 year break and improve your rank. Then, you can get easily any old IIT.

Also, it depends upon on your JEE Mains marks and cut-off of the respective college.

Top ranked NITs for M.Tech are mentioned below:
1. NIT Trichy.
2. NIT Surathkal.
3. NIT Warangal. Regards.

Depends on your category.
For general category, at least aim for 150+. The more you get above it, more are your chances. For OBC and SC/ST, it is a bit low.

Actually, the fee of all NITs is not same even semester fee of NIT-Agartala is not same as the hotstel fee of all semsters. It is 18000 (including mess) and tuition fee varies according to the semester, but approximately INR 70,000 per semester (after batch 2015_19) because the policy of MHRD has been changed from 2016, but if your family income is less than 1.5 lakhs then you need to pay the half of tuition fee.

I can tell you about SRM University. It has 100% placement in CSE department. Many big firms recruit students from SRM University like Cisco, Amazon and E&Y.
You can check the placement records here.
http://www.srmuniv.ac.in/cse/placements/2016-2017
http://www.srmuniv.ac.in/cse/placements/2015-2016

Colleges accepting IIT JAM 2018 scores/ranks for admission are mentioned at the link provided below. Candidates are informed of the total number of seats available for different M.Sc courses offered at all participating institutes. A total of 15 IITs, 16 NITs, and two Centrally Funded Technical Institutes (CFTIs) accept JAM scores for M.Sc admissions:
https://www.shiksha.com/science/articles/colleges-accepting-iit-jam-scores-ranks-for-admission-blogId-13971

If you are getting old NITs with decent branch than go for it, else AIT is a way better option.