MCA

No, you have to study first year too for MCA.
Good luck

Yes, you can apply for government jobs with your graduation and post-graduation.
Good luck.

Some of the colleges you may look for:
University of Mumbai-Fort, Mumbai
VJTI Mumbai - Veermata Jijabai Technological Institute- Matunga East, Mumbai
Amity University, Mumbai- Panel, Mumbai
NMITD - Navinchandra Mehta Institute of Technology and Development-Prabhadevi, Mumbai
The L F College of Engineerin

Yes, if you have completed your graduation in B.Sc, you can surely apply for MCA.

As you have completed B.Sc in Mathematics, looking the future demand and scope of the course and career better opt for AI courses.

You can visit this link
https://www.shiksha.com/resources/college-comparison.

Yes, IITM is a decent college and have all the facilities like modern labs, library, Wi-Fi campus, good infrastructure as well as good faculty team. Good luck.

Yes, any Science graduate including BCA can apply for MCA lateral entry in Pune University. Good luck.

If you have done BCA, then try for MCA (L. E). It saves your one year because first year of MCA has the same syllabus which you have studied in your BCA course. Good luck.

Yes, if you have 60% aggregate in graduation, then you can apply for MCA course. Good luck.