Mechanical Engineering

Yes, it is. But you can also try IITs. I have seen IIT students involve more in research work than any other universities. You can also try VIT, Vellore for updated syllabus and better research work on Mechanical engineering.
Hope this answer will help you.
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Yes, it is actually the best option for engineering.

Both, AMIE and IME section A&B are recognized equivalent to B.Tech by MHRD. AMIE is the most accepted professional and non-formal degree course in engineering.

If you have wish to go for higher studies, then you can join B.Tech through lateral entry. You can do PG Diploma courses provided by different Reputed FTI (foreman Training Institutes) under Government of India.
FTI - Bangalore is a good institute providing such courses.
You can go for CAM/CAD training.

BITS - Pilani is the best non IIT/NIT University for B.Tech in mechanical engineering with placement records equivalent to legacy IITs.

Yes, you can get easily in Canada or UK.

NIBS is the best distance learning center for mechanical engineering.

USA is the best without a doubt. Canada and Australia are also preferred in terms of affordability, RnD and job opportunities.