Distance

In India, MBBS is only available in regular mode. Because One cannot acquire medical skills through distance learning mode. So Its not available in distance or part time mode.

If you are an experienced IT professional MCA by distance education will not be a handicap. But if you are a fresher then it may create few challenges and HR may try to hire you at lower salary in comparison to full time MCA candidates.

For distance education location of institute does not matter. Most of distance learning MBA are conducted on internet now. You can choose between various MBA based on your needs. I will recommend that you evaluate IGNOU for MBA. If you are an engineer BITS Pilani offers good options for MBA such as

Yeah, you can work with them

You can opt for MDS. But distance learning mode is not available for MDS. You need to persue it as full time. Instead you can do some certificate courses.

Yes, there are institutes which provide distance learning and collaborative programs.
5 - 6 programs under BITS Pilani have been mentioned on their website too.
Even IIT Bombay has its Distance learning program, details of which are available on their website - http://www1.iitb.ac.in/~cep/about/index.

You may opt MBA international business from Pondicherry University. this is a central University and has jurisdiction over all over India if you are looking for knowledge, you may opt IGNOU if you are from Delhi and Kolkata you may opt part time MBA from IIFT

government has banned all distance learning engineering programme and such degrees are either fake or invalid