NIT Surathkal - National Institute of Technology

No, you can't apply through MAT score. CAT score (even low) is required.

In NITs, the admission is based on the merit list or rank of exam NIMCET.
You can write exam after completing BCA / BSc course.
The exam is for 1200 marks and to get admission in NIT Trichy you should score at least 600/1200 and for NITK Surathkal 600/1200.(These marks are for general category).
They vary based on the categories.

For NIT-K, I would say a minimum of 50-55% to get selected for the 2nd round. However, note that, GMAT or CAT scores are weighted at 70% in calculating the overall score for the final selection. So, problem with the low score is, it will haunt you even after getting a call for GD and PI based on your application. Higher the marks, better is the chance to get a seat. Else, for GD & PI, 50-55% should be good enough.
All the best.

It is a very low score. Try to score good marks in GMAT.

This depends on college to college. Usually by February 1st week most of the colleges are expected to call for application. Start shortlisting your colleges and check their respective websites. Also, keep following Shiksha for regular updates on entrance process.

You can apply for NITK-SOM (Good ROI), JKSHIM-Karkala and Manipal-SOM (Both are old and reputed, placement is better than average), Sahyadri Mangalore or AIMIT (Both are good at academics and extra-curricular activities, placement is decent).

Usually, it is CAT and not Joint Management entrance test. However, please check each IIT's profile in Shiksha website for entrance details.

BITS-Pilani would be the better choice if the interest of engineering discipline is towards computer science/ electronics as it has got stupendous placements in these domains. However, this doesn't imply that NIT-Surathkal has a poor placement record in CS/EC. Its just that, BITS has got an upper edge in these domains than NIT-S. If you want to go for core branches like Civil/Mech., go for NIT-S. The campus life of BITS is awesome whereas NIT-S has its own beach where students can tranquilize. Both the institutions are good and have their own pros and cons. I suggest you to select one as per your branch and preferences.
All the best!

Selection to national institutes is based on the cut-off marks scored by a candidate in JEE Mains and the structure of option entry. For the residents of Karnataka, seats may be available if NIT-K is given the first option in the entry. It also depends on the branch selected by the student.