Colleges in Rajasthan

Yes, it is worth to study at BITS, Pilani because it is an international university and it is recognized world-wide. You can get better placements and a lot of knowledge with a great exposure.
NITs are better in terms of fee and education but not better than BITS. NIT is a government college.

You cannot get admission in a government engineering college. Try for private institutes.

No, BITS, Pilani doesn't offer mechatronics.

For the first year, I'm not sure about the fee concession. Maybe your BITSAT score can be considered. If you are a board topper, then you will a get full fee waiver.
But from the next semester onwards, if your CGPA is above 9 or 9.5, then you will get mess fee and hostel fee waiver. But this depends on your CGPA and the administration.
If you get the above CGPA, there is an option to change your course also.
All the best.

You can get the fee structure on the official website. Yes, there is fee concession on the basis of class 12th score.

The marks scored in BITSAT doesn't matter for scholarship in Indian campuses of BITS-Pilani. For Dubai Campus, scoring above 300 would be sufficient for 75% scholarship.
For Pilani, Hyderabad and Goa campus merit and merit cum scholarships are available. Merit scholarship is available to top 15 students of campus while MCN is applicable to those whose parents income is less than 7 LPA. Also, for MCN Scholarship, CGPA has to be greater than 8.5 for 80%, between 7.5 and 8.5 for 40% and greater than 6 for 25%.

Hi Rajat,
There are only two ways for getting into BITS-Pilani, one is you need to score 300+ in BITSAT, or you need to get the highest marks in your boards from your state.
All the best.

Aayojan School of Architecture would be an option, other than that you can check the Council of Architecture approved colleges in the link - https://www.coa.gov.in/show_img.php?fid=216