Distance

The most common choices for you are:
Chartered Accountant (CA)
Masters of Commerce (M.Com)
Masters of Business Administration (MBA in Finance)
Certified Management Accountant (CMA)
Certified Public Accounting (CPA)
Association of Chartered Certified Accountants (ACCA)
Business Accounting and Taxation

No universities can provide Masters through distance education in culinary arts, this is a practical course that cannot be learned through books or online. You need to attend class do practical and then get certified.

Yes, it is possible you may apply to TISS (alone MPhil is not available you have to do integrated M Phil and PhD) and you may search other institutes too.
Please also note that institutes may ask UGC DEB approval and AICTE approval to your distance MBA program.

You can refer the link: https://www.shiksha.com/mba/colleges/distance-correspondence-mba-in-human-resources-colleges-india

Yes, of course you can and if you have good cat score than your academic record and overall personality, communication skills and knowledge will be main point to get the seat in good B-school.

Strategic management is not a full fledged specialisation of MBA. which university is offering this?
For distance learning, it is a useful only if you are already in a operations role and your organisation wants you to move into role where you have to adapt and use different strategic tool which you

SCMHRD from Pune is good, even Welingar is good. They assist you for placements too if you want it. They are reputed courses and can be done from far distance.