B.Tech Computer Science and Engineering

For the general category students, it will be difficult for to get into that but yes if you belong to SC category then you might be able to get that.

Yes, you can apply for the admission through management quota. If you come under their eligibility criteria, then definitely you will get admission.

It is INR 1,65,000 for all branches irrespective of rank.

There is no any admission charge, you have to pay annual fee.

In class 12th, you must have to score at least 75% marks to get admission into NIT's or IIIT's. It is just a qualification criteria, ranks are not determined by the board marks. Minimum percentage of 75 should be there to join in these institutions.

amity in Mumbai is excellent

Sinhgad is the best option as compared to Shri Ram Institute. This Sinhgad has good teaching and non-teaching faculty members. Also, the placement package given by the college is also above 3 LPA for a fresher.

CSE from VIT should be taken.

Im really not sure about this,as far as I know it may be in between 8.5k - 24.5k for OC/general category in EAMCET(considering all the four branches).
As you know, this ranking preference changes according to gender and caste grade, it fluctuates every academic year.
If your rank satisfies here, you will get admission.