CBSE

280 is a decent score. State your AIR or use Shiksha college predictor service.

Rarely, there is a chance for you to get into such a prestigious college with that percentile.
But don't worry. Try other colleges. You will surely make your life better, there too.

Your brother can opt for Improvement in the subjects he recieved less marks. For that refer to the CBSE Website at https://www.cbse.nic.in
Also refer to https://www.toppr.com/bytes/cbse-improvement-examination/for more information. Other choices your brother has are that he can pursue a UG course at local college and prepare for a UPSC Exam simultaneously.

Honestly, no one leaves CSE branch seat. If there's any seat left, students from other branches will take it. As you haven't given JEE, there's a very little chance to get admission in a branch of your own choice.
But yes, last year, at the end, admission to Thapar was open to all. You could get a seat in the branches available at that time.
Usually, Electrical, Chemical, Mechanical, Civil have some seats left.
Rest, visit www.thapar.edu for more information.

Yes, definitely, you will got an engineering in any state, as per your JEE marks.

You are not likely to get a government college in India. Hence, you'll need to get either into big private universities or into local engineering institutes. Check the best colleges in PUNE for Engineering or either apply for MANIPAL, VIT, AMITY, AMRITA or LPU for distance education. NIMMS also if you can afford.

If you have any quota, then maybe you can get admission.
It also depends on this year's cut-off.

Firstly, you can go through the chapters which you thought you can easily deal with it and do more questions of it.
Then step by step, you can go to slightly tougher chapters.
It will increase your confidence.

No, because in science stream, PCM are the main subject and also you can not do chemistry or physics honours. Because they also require maths. You may go for hons in physical education or hons in English or other humanities stream subject courses.