Degree

Yes, you can get admission in AU but you need to get a good rank in AUCET exam.
Distance mode is not a criterion.

Singhania University has now been granted the approval to conduct AME course as a regular Bachelor's degree. Hence they have started offering a Bachelor's degree i.e. B.Sc in AME. Only those candidates who have qualified TSICET-2018 and secured 50% (for OC) and
45% (for others) in aggregate marks degree or its equivalent examination can apply for this. Hence your degree qualification should be acceptable and you may get admission provided you score the minimum required marks in the entrance exam.

No, backlogs need to be cleared before any higher study. Clear your backlogs first because most colleges will refuse to give you admission until you do so.

This depends on your area of interest. If you want to continue your career in the pharma field then you should do MBA in pharma itself. If not, there are many other fields you can go for.

The college you mentioned provides the said course.
https://www.shiksha.com/humanities-social-sciences/political-science/course/master-of-labour-welfare-mlw-veer-narmad-south-gujarat-university-surat-171796

I think you can. Your PG degree must have mentioned somewhere that you are a Bachelor + Masters. So, you should be able to apply.

Yes. But, sometimes, colleges require that you submit a self-declaration stating that you will clear. Your backlog before a set time or your admission will be cancelled.

MBA in India is 2 years and is considered a Masters Degree. In foreign nations however, it can range from 1 year to 3 years. You will have to check each college individually. Also, not all courses are considered a degree. You will again have to confirm that for each college.

I prefer you to go in NID and apply for it and view the previous year admission list of students. You will be considered like a B.Tech student only, so go for it.