Colleges in Mangalore

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.

https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
Please go to the college predictor link I have shared here to know about the college you can get.
good luck

KCET 2018 College Predictor
Use your KCET 2018 rank to see where you can get admission at the link provided below:
See the courses, colleges & rounds you may crack
The result is based on admissions data of KCET 2017
Get resources & insights on colleges & courses
https://www.shiksha.com/b-tech/resources/kcet-college-predictor.

NIT do not have lateral entry system. In NIT, Diploma is considered as equivalent to class 12th, if you want to get admission in NIT, you have to appear in JEE Mains and score a good rank.