B.Tech Mechanical Engineering

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.

Hello
Yes!
If you are interested in pursuing Agribusiness Management at IIM, you can apply for that, after completing B.Tech! IIMs Agribusiness Management is two years programme.

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%