B.Tech Mechanical Engineering

After B.Tech, you can do M.Tech in whatever field interests you. Therefore, it is up to you.

Kindly visit this link:
https://www.shiksha.com/b-tech/resources/mhcet-college-predictor

In most of the reputed colleges, B.Tech lateral admissions are done through the entrance test. Hence, you have to apply for respective college's lateral entrance test first. Following are a few colleges offering B.Tech lateral admissions:
1. DTU, Delhi
2. Thapar Institute of Engineering and Technology, Patiala
3. PTU, Jalandhar
4. College of Engineering Pune,
5. Anna University, Chennai.

Yes, definitely you can do MBA in logistics and supply chain even without an experience. Since you are an engineer, it will be easier for you to cope up with the course.

Yes, you can get admission in this college directly. you dont need to clear Jee Mains. There is no selection criteria of admission in this college.

I can't assure you 100%, but If you wrote JEE exam and score good percentage you can.
With your quota, you can apply directly in college you can't attend JoSSA, unless you write JEE (main & advanced).

I am attaching a link where you will get all the related information to get admission in University of Hyderabad:
https://www.shiksha.com/university/university-of-hyderabad-uoh-23069/admission

For College of Engineering, Anna University, Chennai the opening marks and closing marks were 194.25 and 200 respectively. For PSG College of Engineering, it is 194.75 to 199.5, Anna University Tiruchirappalli it was 138 to 195.5. You have scope for other branches.

In current scenario, the placements are not very good for Mechanical Engineering, but if you are getting a government college, then there are chances you might get placed with a good package.