Rajasthan

For fashion designing: https://design.shiksha.com/fashion-textile-designing-courses-in-jaipur-categorypage-13-69-1-0-0-109-121-2-0-none-1-0
For beautician course: http://www.indiacom.com/yellow-pages/beauty-training-school/jaipur/

You can ask the institute to provide a data on the same

if you have control over yourself to stay focused in the drop year for JEE Mains preparation, then go for it. otherwise, take admission.

After taking up NATA you can enter an Architecture college, you need to take up Medical entrance exam for Government Medical College.

Dear don't go for coachings they will kill Your real self

Hi Ayush,
If possible go through the alumni speak section to get an idea what the alumni thinks about the institute. However, we can help you with the parameters that you can keep in mind while choosing an institute. You can consider the below parameters :
1) Faculty,
2) Affiliation / Recognition
3) Course Material,
4) Quality Education,
5) Fees
6) Infrastructure
7) Success Rate.
8) Placement
9) Top recruiting companies.
Hope this information helps. Please feel free to revert.
Wish you luck.

You can refer this link for college information: https://www.shiksha.com/Sgvude-Suresh-Gyan-Vihar-University-Distance-Education-Sgvude-Jaipur-institute-college-listingoverviewtab-47867

I don't think that you should go Kota for JEE Mains preparation because in Delhi only there are so many institutes. I would like to tell you one thing that everything depends on your efforts and your self study. Any of the coaching is not responsible for your selection because faculties show you the path, they guide you. now it depends on you that you have to walk on that path or not.
So the conclusion is join any of the best institute in Delhi only and give your complete effort you will easily crack out your JEE.