B.Tech Mechanical Engineering

Yes, you can do. The eligibility criteria of M.Sc Mathematics (Hons) from LPU are as under. Eligibility Criteria (Qualifying Exam) :- Pass with 50% aggregate marks in Bachelor's degree (any discipline) (with Mathematics as a subject) or B.Tech or BE or B.Sc (Engineering) or equivalent, subject to qualifying LPU-NEST.

Yes, it is possible. M.Des is a great course for the creative types and your position and salary would boost after the course. Some great university like UPES, Sharda University conducts the programme. So, you can apply for it. But it is necessary that you have at least 50% at graduation level.

Yes, you can get lateral entry in mechanical engineering.
For more information, click the links:
https://www.gbpuat.ac.in/academics/admissions/index.html
https://www.shiksha.com/college/govind-ballabh-pant-engineering-college-gbpec-uttarakhand-other-24262

Yes, you can take admission

Sorry, to say this, you are not eligible for that, because it's base cutoff is 75%

SAT is accepted by the following Indian universities as well. The number of Indian universities accepting SAT scores from Indian students in India is increasing. The universities in India include
- Ashoka
- Manipal
- Azim Premji University
- Ahmedabad
- Bennett
- BML Munjal
-FLAME
- Manav Rachna
- SRM
- NMIMS
- GD Goenka
- Krea
- MS Ramaiah
- Shiv Nadar
- Srishti
- GITAM
- Sri Sri University
- World University of Design
etc.

For NIT, Warangal you should have at least 96 to 97 percentile.
Better luck for April attempt.

Hello Kshitij,
Almost all colleges provide lateral admission bar few like NIT's or IIT's or few deemed to be universities. DTE of tech education of your state will have this information on it's website. You can even download it in many states. Best wishes to you.