M.Tech

Check out the reviews below and select the course :
https://www.shiksha.com/university/manipal-university-manipal-academy-of-higher-education-30822/reviews

Yes, you can get admission to MIT WPU, it accepts all type of entrance exams CAT MAT CMAT MH-MBA CET just that you need to hurry, as the late you approach the cut off will go higher .
But, the good thing is MIT College has become MIT World Peace University now. So, intake of students is increased by college, so there are more chances.
You can also look for Balaji in Indira College.

All courses abroad or specialized one. You can do PG Diploma or PG in:
Construction management
Structural engineering
Geo technical engineering
Go through few university website and see the course options.

Yes, you can take admission into M.Tech course without GATE score card. You can take admission in state technical university affiliated colleges by appearing for an entrance exam conducted every year by respective state university/colleges etc.

Clear GATE exam, apply for the college/PSU if you clear the cutoff you will receive call for the same.

Yes, it is possible, if you work hard and get a decent score in GATE. I think you can wait and you may get a chance in any of the good NITs.

Deeply clarify what actually you want to make your career. If it's maths, go for M.Tech. It's a higher qualification than any M.Sc. It gives easy placement and better job prospects.

Yes, it is possible that you do B.Tech in some college and M.Tech in some other college for that you have to give the GATE exam. The Maharashtra MTech 2019 cutoff is not yet declared. Generally the cutoff remains more or less same . COEP is a good institute to pursue your MTech and get placed. I hope you got your answer. IF not let me know in comments section. All the best.

Yes, you can appear for GATE more than twice. ECE us a good branch but their scope in India is less if you want a job after M.Tech or B.Tech. Usually, doing M.Tech in ECE is good than only B.Tech. Moreover after M.Tech many companies doors are automatically open for you. Also some PSU may give admission based on that and the score you achieved.
I hope it helps.