Colleges in Rajasthan

Hey Tarun,
SBI provides scholar loan (education loan) to the students securing admission in country's best engineering colleges at an interest rate of 10.75% per annum irrespective of the loan amount. This loan can be availed from the nearest SBI branch of that campus.
A scholar loan has a maximum limit of 20 lakhs (without security) in some selected institutions of India. BITS-Pilani is one of them.
An education loan provides only 4 lakhs (without security) and 7.5 lakhs (with security) whereas a scholar loan provides a higher amount.

Here is the list of colleges offering M.Tech in automobile engineering:
NIMS University (NIMS), Jaipur
Singhania University (SU, Jhunjhunu)
OPJS University
Dr K N Modi University (DKNMU, Tonk)
Jagan Nath University, Jaipur
Mewar University

The placements are not so dedicated. Hence, it is not recommended.
You can check the below mentioned link:
https://www.shiksha.com/mba/course/mba-dual-specialization-pacific-institute-of-management-pim-udaipur-259902

Yes, it is valid as per UGC act.

You should not judge a university by the report of news channel. It provide degree as per UGC norms.

Yes, as per UGC act it is a recognized university.

Yes, LLM course is valid.

Yes, you can do it as it is a recognized university.

No admission is offered on the basis of sports quota or any other quota at BITS. Admission is totally on the basis of merit and through BITSAT exam only.

The cutoff for mechanical engineering for BITS, Pilani is 346 and in BITS, Goa campus it is 322 for mechanical and 316 for Hyderabad.