Maharashtra

Please refer to http://joinindianarmy.nic.in/how-to-join.htm
You can try for 10+2 Technical Entry scheme.
Age - 16-19 Years
Qualifications - 10+2 (70% marks with PCM)
College of Military Engineering is a premier technical and tactical training institution of the Indian Army Corps of Engineers which includes Combat Engineers, Military Engineering Service, Border Roads Organisation and Survey.

1. University of Mumbai.
2. ITM - Institute of Health Sciences.
3. Mahatma Gandhi University – Mumbai.
4. MGM University of Health Sciences.

Although, transfers in the same university transfers do happen but it is not a very common scenario in India. Moreover, you need to get a transfer certificate from the college which at times very difficult to get, so I would suggest you talk to your college officials and if they agree to give you TC then talk to the college you want to get into once you get clearance from both the sides.I don't think you will face any problem in getting transferred.

Yes, it is approved, you can see the list of many others on UGC's official site.

The exact syllabus cannot be determined until after the paper, however you can get the general idea of the syllabus along:
• Macroeconomics
• Mathematics
• Micro
• Econometric
• Stats
• Probability
• Game theory