NIT

Your NTA score is 20.8 means your estimate rank is above 5 Lacs, so it's very unlikely for you to get admission in NIT, until you are from ST category, but you can go for a private college that accepts on the basis of 12th class or management quota.

Yes, Mechatronics is good in VIT Vellore as it involves a modern curriculum, strong labs and research, good faculty, and industry exposure. Mechatromics is equally good, I'm IIT, NIT with slight differences.

With a 73 percentile in the JEE Main and belonging to the ST category, admission to NITs for B.Arch is unlikely for top-tier NITs, but there's a chance for some NITs and other good architecture colleges. You should explore other colleges and consider participating in the JoSAA/CSAB counseling process. 

Your OBC NCL rank 7000 in JEE Mains puts you in a favorable position for admission in the following NITs: NIT Calicut, NIT Jamshedpur, NIT kurukshetra and NIT Silchar. Another good IIIT is IIIT Sri City, Chittoor which is on par with IIIT Nagpur. Your chances at NIT Jamshedpur are good.

No, You can not get admission for B.Tech (any branch) in NIT with 52.34 percentile. It's because the cutoffs of NITs are set between 85-90 percentile for reserved categories and above 95 percentile for general category. However, you can try for State Government Colleges and Private Universities/Colleges that either have their own state level exams or lower cutoffs respectively.

Yes, you can get admission in engineering colleges like IIIT or NIT, but it depends on your entrance exam performance.

For M.Sc Mathematics admissions in NITs, the entrance exam is typically IIT JAM, not JEE Mains.

You particularly have to give IIT JAM for M.Sc.