Delhi

Instead of distance, I would rather say that you should pursue a full-time course as a full-time course will gain you lots more knowledge and you'll be able to secure good placements.
Some good colleges are:
Great Lakes, Chennai
Jain University, Bangalore
Rajagiri Centre for Business Studies, Kochi
Pune University
NIT-Agartala
LIBA, Chennai
ISB, Hyderabad.

MCA has a wide variety of applications and it's scope in India is also really good, because India is moving towards development in IT sector and Indian IT industry prefer B.Tech CSE and MCA for application development and maintenance roles. So, it's a good choice to go for MCA.

Yes, you are eligible.

Hi Ayushi, Please refer the link below:
https://www.shiksha.com/mba/exams-pc-101
https://www.shiksha.com/mba/articles/5-reasons-mba-aspirants-can-do-without-cat-blogId-7087.

Hi Arjun,
Please refer to the link below:
https://www.icsi.edu/media/webmodules/aboutus/courses/CS_Course_01042014.htm.

Yes, off course you can convert it.
All the best for your bright future
Work hard and smart to get success. Thank you.

Below are the institute's names.
1. CAAS
2. The Analytics
3. Classic IAS Academy
4. Chanakya IAS Academy
5. Jain's IAS Study Circle
6. Pragati Center.

Chances of converting is high, but you need to perform well in GD, PI and extempore. These will be deciding factors not only in FMS-Delhi, but also in other IIM conversions.
Weightage of composite score:
CAT - 60%
GD - 10%
PI - 15%
Extempore - 5%
Class 10th - 5%
Class 12th - 5%.