Mechanical Engineering

You can pursue a Masters in Engineering Course - ME / M.Tech. Programme with specialization in Mechanical Engineering (Two Year/ Four Semester Duration Course).

Yes, it depends. Your college must be either UGC or AICTE approved. Your academic record, work experience, etc. matters a lot.

Kindly predict your college in this college predictor. JEE Main rank predictor is an online tool which predicts the candidates rank range and probable colleges based on the difficulty level of the examination, average marks scored in JEE Main and seats available in IITs, NITs, GFTIs, and other SFTIs. The rank predicted might not be accurate but it certainly does give an idea to the aspirant. Know your college admission chances based on your JEE Main Rank, Home State, Caste, Gender, etc. Here https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

Sikkim Manipal University offers distance learning programs, but it is recommended not to do B.Tech in distance learning. Rather, you can opt for AMIE i.e, B.Tech from correspondence.

You should join mechanical engineering after mechanical Diploma. You will get lots of things to learn. And mechanical has good scope ahead. Good luck.

You can't get admission in direct second year if you don't have Diploma. So, if you have Diploma you can fill for counseling otherwise you have to start from first year itself.
Good luck.

Both branches are very good, but it all depend upon your work interests, but going with ME there are more advantage than chemical engineering.Go through the link:
https://www.shiksha.com/b-tech/articles/what-mechanical-engineers-learn-blogId-3657
https://www.shiksha.com/careers/chemical-engineer-134.

As a mechanical engineer I would suggest you to go for chemical engineering, if you are not good in drawing, because in ME you have to face engineering drawing, mechanical engineering drawing, and basic engineering drawing.
You will face very difficult in tackling these subjects
Good luck.