Mechanical Engineering

Automobile engineering is good.

You can check the below mentioned link:
http://www.bits-pilani.ac.in/university/wilp/AdmissionFeesStructure

Presidency university does not offer BE course, for course structure of other courses , follow the link below :
http://presiuniv.ac.in/web/fee notice.pdf

What you mean by abroad? You must be specific with your country choice. If you haven't made a choice yet, you can check websites like QS World Ranking, Economic Times, Times Higher Education, etc. for the university rankings worldwide. You will have the idea. You need to take IELTS or TOEFL as well to apply.

Lovely Professional University is good for supply chain management. It provide good placement and career opportunity.

Yes, it is a good decision. After completing diploma, you can opt for lateral entry in B.Tech on the percentage basis or through LEET examination.

Both the colleges are equally good but to know the exact scenario for faculty and placements, get in touch with the current students through social media or in person.

Yes, there are many colleges in Bangalore for lateral entry admissions.

Keep the branch as your first priority then college, don't mix it because eventually the branch decides your career.