Nepal

Yes, surely you can try for it.

Hello Manish, there are 2-3 entrance tests that you ought to give to secure admission in India. One of them is JEE Mains, then JEE Advance. The fees of any engineering college or university is not more than the figures you described. As far as good colleges for Mechanical Engineering are concerned, the link below has the list of them-
https://www.shiksha.com/b-tech/ranking/top-mechanical-engineering-colleges-in-india/48-2-0-0-0

Hello! Sorry for late reply. The fee structure is nearly 1.60 lakh per annum.

Hi,
Please check UGC website for university recognition, if it is listed you need not worry. However you must still write to the universities you want to apply to and check acceptance of your degree, every university has different admission requirements.

Sanzok, you can definitely apply for admission in Diploma. Check the link below:
http://www.lpu.in/international/
Open the link and apply from here or call on the no. given in the link.

Hi Navin,
As far as FMS is concerned: Yes, Nepalese can definitely pursue MBA full time course at the institution.
In fact there have been a few Nepalese who are currently pursuing MBA from FMS. You no longer have to have a GMAT score for being considered for admission into FMS. Please contact your embassy for detailed admission guidelines and procedure to be followed.