GATE

Dear,
M.Tech courses in India is only available to those who had B.E /B.Tech. So, instead of thinking of M.Tech, you can proceed for some better courses which you can get easily with good GATE score.

Admission is through counselling based on rank obtained in Tamil Nadu Common Entrance Test (TANCET) and not gate , for more details check out the link below :
https://www.shiksha.com/architecture-planning/course/m-arch-landscape-architecture-school-of-architecture-and-planning-anna-university-guindy-chennai-203114

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.

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.

It's just from my own experience that I am saying. In Delhi, I am aware if one good institute which is Made Easy. It is good institute which covers all the topics and they have a very good teaching faculty members. They are even good at doubt clearing and moreover their notes are very good. I have seen their notes and it good. If you just practice that you will be confident enough to crack GATE.

Only 3 months left. That's very less time. Still don't lose hope. I am not sure what is your branch so I cant be specific. I can be general. So make your concepts clear in the first month and don't try to cover each and every subject. Just focus on the subject which is maximum marks and also the subject which are the easiest one. Just focus on these subjects and solve previous year papers as much as you can.