GATE

All IITs use Graduate Aptitude Test in Engineering GATE score for admissions to various post-graduate education programs (e. g. Master of Engineering, Master of Technology, Doctor of Philosophy). There is no direct admission in higher studies at IITs except for sponsored candidates.

Every IIT has it's own cutoff for admission in different branches.
So, I suggest you to score more as much as you can.
Do hard work and try to solve the paper correctly and less mistakes.
Just check the cut- off for the IIT
You can see the cut-off that is so high
Just concentrate on your studies and focus.

You must specify which course you want to pursue and the stream in which you are interested.

Yes, there are so many private universities in Mumbai and other cities also, where you can take admission in M.E. without GATE exam

Yes, you are eligible to get admission in LPU.

I have heard of the universities like MIT, Berkeley and Princeton.

Yes, AMIE degree holders can apply for GATE exam. In case a student is expecting to complete the qualifying criteria in 2020 or later as mentioned in the table, then he/she has to upload their Provisional Certificate Letter or a copy of marks card for Section A of AMIE, etc. ,because you have not completed your AMIE so far than you can submit AMIE Section-A mark sheet while applying for GATE exam.