KCET

There is a website which predicts rank and even in shiksha website, you can predict your CET rank which will be not so accurate.
In https://shiksha.com, there is a college predictor. Check it out.

Choosing mechanical is one of the best decisions as compared to other branches. You can go for RV, BMS or Dayanand Sagar or any other college.

You can't get admission in a top college with this rank. If you are from SC or ST category, you can get admission in a good college but not the top ones.

Refer the following colleges:
NIT (Surathkal)
MIT (Udupi)
NIE (Mysore)
RVCE
BMSCE
MSRIT
Siddaganga Institute of Technology (Tumkur)
PES
UVCE
BIT

Yes, you can apply. To appear for KCET, the eligibility is as follows:
1) Candidates should complete 10+2 from a recognized board under Karnataka state.
2) Engineering candidates have must studied PCM with English.
3) Candidates must have scored 50%.
4) Candidates must be the resident of Karnataka state.
5) The minimum age limit is 17 years.

Yes, as most of the colleges in Karnataka will check for appreance in KCET or ComedK exam. You need to qualify any of these entrance tests for admission.