Colleges in Mangalore

It depends on your branch of engineering.
CS - 2000
Civil - 30000
Mechanical - 40000
Please do share your interested branch of engineering so that I can help you in detail.

Yes. Its there. Here are the list of colleges:
1. East west
2. Sambram institute of management studies
3. Rr institutions.

For NITK, please compare with the cut-offs of the last year.
https://www.shiksha.com/college/national-institute-of-technology-surathkal-mangalore-24187/admission#admission

If you are eligible to get a seat in Electronics and Communications, then I would suggest you take it over Electrical and Electronics in NITK. As EC is a better branch in terms of faculty and placements, then EEE in NITK.

Board marks doesn't matter, but depends only on your rank. Please compare with the cut-offs of the last year for NITK.
https://www.shiksha.com/college/national-institute-of-technology-surathkal-mangalore-24187/admission#admission

This is usually possible for private colleges. I haven't seen such cases as of now. I think, you will have to write JEE Mains and get into first year like anyone else.

Please compare with the cut-offs of the last year for NITK.
https://www.shiksha.com/college/national-institute-of-technology-surathkal-mangalore-24187/admission#admission

Go for NIT-Warangal, placements are good for ECE students.

I also had almost same rank as you 1578 (AIR_OBC). I will suggest you to select NITK Surathkal IT. Even CSE at Allahabad is also a very good option if you are from UP. So if you want to stay near to home then go for CSE at NIT Allahabad otherwise come to NITK, Surathkal in IT.

Yes, you can. Eligibility for MBA is any graduate with minimum 50% marks; that's it for most of the colleges.
But some colleges that provide PGDM will check your record. They may not give you admission for PGDM but you can get admission for MBA where admission takes place directly on the basis of your entrance test performance.