Engineering Entrance Exam

The college predictor does't have all the colleges which accepts JEE Main scores. But you have a chance of getting good state engineering colleges. You can look for state engineering colleges in your home state.

No, it offers admission to other students as well. But, it depends upon students 10+2 percentage and JEE rank. For a few details:
http://www.cuchd.in/admissions/.

If you aren't of general category then you are qualified other wise you have to look for private college if don't want to drop a year. You have to qualify for one entrance exam in getting the admission. There are many college which admission on basis of 12 % like LPU, UPES, ICFAI and many other priv

Yes, you will get a cut-off probably around 100.

Hello,
I hope you are doing well. You can check the list of college including NITs and IIITs, you can get according to your marks and rank here.
Use Shiksha JEE College Predictor:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=cconnect&utm_medium=CR_JEECP&utm_campaign

Firstly, I would like to appreciate your mindset for aiming 300 marks in JEE main exam. Yes, you can definitely get such mark but all you have to do is to get all your concept clear and keep on hard working continuously and by doing this you can achieve good marks.

Top Engineering Colleges in India Accepting BITSAT are mentioned at the link provided below:
The ranking is based on rankings published by India Today, The Week, and National Institute Ranking Framework (MHRD). https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-india-accepting-bitsat/

It may varies from year to year.
It is released after results.

There is no date announced yet for the result if PESSAT. But they will announce the result soon.

It is based upon your interest to choose between various universities.
I suggest Presidency. Vignan is also not bad.