Mechanical Engineering

Yes, you can do, But colleges have some distance restriction. Like Madras Institute Technology is saying that, A candidate seeking admissions to part-time mode of study shall have his/her permanent place of work within a distance of 90 km from the study campus of the institution. The following are the other colleges offering the M.E/M.Tech part time programme. For details please see the concerned college website.
1. PSG College of Technology, Coimbatore
2. Delhi Technological University.

The objective of these Diploma courses is to provide a comprehensive understanding of various aspects of Mechanical engineering and skills development in Plastic Technology, CAD, Mechanics, Material Science, Automobile technology, and Manufacturing and Production. Diploma in mechanical engineering have good career prospects:
Railways
BHEL
BSNL
Coal India
NTPC
BEL
ONGC
DRDO
SAIL
NACL
Junior Engineer
Assistant Engineer
Apprentice
Visit: https://www.jnujaipur.ac.in/.

Yes, you can get admission in Delhi University without giving any entrance exam.
They directly give admission on the basis of marks you scored in 12th standard.

You have to complete it first and then apply for the same.

This depends upon the Diploma but the value of a degree and Diploma is different.

Hello Rushikesh,
If you want to make your career in B.Tech (mechanical engineering) then you should choose Science PCM stream in your 11th and 12th class. PCM means physics, chemistry and mathematics. Study carefully and with attentive attitude.

Dear Ramanath Mishra,
As per my knowledge, you can not get admission anywhere by showing the degree from EIMSR because as per the data available on the internet this particular institute is completely fake there are many people complaining about this institute, though I do not have any valid information about this EIMSR. So I would recommend you to cross-verify your BE (Mechanical Degree).

Yes sure. All you need to do is cracking and ranking very good in GATE(cse). But I will suggest that it will be better if you can continue with mechanical engineering in M. E. because your plan of suddenly switching your stream from core to noncore might backfire .

I don't think any college would offer such change in stream. But even if it does, (you might get IT but not CSE) you should be well prepared about the pre requisite. Learn C for starting because they already know C language as it was in their first year course. Moreover, their subjects might seem difficult to you as you've read mechanical engineering before.