IIT

JEE Advanced is conducted mainly for the IITs to admit students. But there are a handful of other National Institute that consider JEE Advanced marks too. They are
1. Indian Institute of Science (IISc), Bangalore
2. Indian Institute of Petroleum and Energy (IIPE), Visakhapatnam
3. Indian Institutes of Science Education and Research (IISER), Bhopal
4. Indian Institutes of Science Education and Research (IISER), Mohali
5. Indian Institutes of Science Education and Research (IISER), Kolkata
6. Indian Institutes of Science Education and Research (IISER), Pune
7. Indian Institutes of Science Education and Research (IISER), Thiruvananthapuram
8. Indian Institute of Space Science and Technology IIST Thiruvananthapuram
9. Rajiv Gandhi Institute of Petroleum Technology (RGIPT), Rae Bareli
While these Institutes are a niche category, they sure do provide good career prospects, but it'll be more research oriented. Hope it helps.

It depends on your level of preparation. Factors to consider? One way to know is to take mock tests. You can even take tests online based just on your 11th syllabus, so take a shot to see where you stand.
If you find that you are good at understanding the concepts easily, have sufficient problem-solving abilities, your preparation level is decent, but you need someone to keep kicking you to keep studying, then Kota might be the place for you!
Kota can't create some entirely new abilities in you in just 1-year time, especially because you will take a few months to just get used to the new life.
If you feel you are that far behind in studies, chances are that Kota will further demotivate you because of the intense competition and stress, so you would have a better chance when you prepare from the beginning at a good coaching class near your home.
My personal choice? I would choose FIITJEE, Delhi. I studied in FIITJEE, Gurgaon during my drop year and it was a good experience. I've also heard good things about the FIITJEE, Delhi batch. Though I've not been to Kota, I've heard pretty awful things about the life there. Lot of pressure, stress, competition. That does bring out the best in some people. But it would definitely not have worked for me. It's also harder to take care of your health when you are living alone for the first time. So, in case you fall ill easily and are not used to living alone, avoid it.