B.Tech

No, you can't get admission for B.Tech. You can do B.Com BBA etc. But, you can do company secretary course, it has bigger scope.

Sponsored (S): The sponsored category applies to relatively lower merit category with high fees. Generally, students who fail to score great in selection test are given direct entry with higher sponsorship charges. Non-Sponsored (NS): It is a high merit category where a student competes and clear the criteria set up by the selection board. The fee structure for the non-sponsored category is less than what is asked in paid seats (Sponsored).

Yes, it doesn't matter. You can also be eligible for GATE Examination. Best wishes for your future.

BE has more scope than B.Sc. You only need a Bachelor's degree or want to go in teaching or some research field, then you should choose B.Sc. And, if you need some more scope and you want to make a big career then you should go for BE.

It depends upon you. There are so many branch in B. Tech. In my opinion, for future purpose, fire and safety is best, and I also prefer for chemical branch.

You are eligible. You can get admission in top private and Government College for B.Tech.

There is no course as FSP in JNU Fine Arts College. The course offered is B.Tech Facilities & Services Planning. To know about the details refer:
https://www.jnafau.ac.in/courses/b-tech-facilities-services-planning/