Distance

Yes, you are eligible. Minimum Master's degree is required with 55% marks.
For more information, click on the link.
https://www.shiksha.com/engineering/course/ph-d-wellborn-group-of-professional-studies-lucknow-206570

Fashion designing is a creative field and you can sharpen your skills under proven and expert teaching staff. Therefore, I am of the opinion distance education can not serve as a replacement of regular formal education particularly in the field of fashion designing. You are advised to seek admission in a good fashion designing institute to learn finer skills of fashion designing.

Yes, this is good only if you are working anywhere with your distance learning and are in your core field.

Hello Sudha,
If you want to take distance education in B Ed, better find out if your state is accepting this. Else, all your efforts will go waste.
However, the best option is to approach college in your place and see if they have afternoon classes. But my advice is to do a PG in engineering. Then you can teach in a government or any private engineering colleges.

IGNOU will be the best option for you.

Yes, you can do but it's not advisable. Generally, there will be 4-6 semesters it differs from college to college. Fee also differs but it may ranges from 20000-60000 per year depending on the college.

Kindly check the below mentioned link-
https://www.shiksha.com/business-management-studies/ranking/top-executive-mba-colleges-in-india/18-2-0-0-0

Actually, dear M.Sc in stats is a wide course related to business and economics. So I will advise you to go for this course on a regular basis. And, in terms of college, you should apply to central universities and to TATA institute also.