Colleges in Mangalore

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.

Yes, you can if you have completed 10+2 from a recognized board.

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!