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First of all I will answer for aeronautical engineering -
The best Aeronautical Engineering colleges in India include the Indian Institute of Space Science and Technology, Punjab Engineering College, Institute of Aeronautical Engineering Hyderabad, Tagore Engineering College, etc. Indian Institute of Space Science and Technology - IIST is ranked 43 by NIRF Ranking 2022 with a NIRF score of 52.37 among the top Engineering colleges in India. The list also includes Punjab Engineering College at 87 and Jain Deemed-to-be University at 115 with NIRF scores of 41.05 & 38.55 respectively. There are around 98 Aeronautical Engineering colleges in India which include 86 Private and 12 government colleges. IIST Thiruvananthapuram offers Aeronautical courses at around INR 1.70 - 2.50 Lacs and has 140 B.Tech seats. MLR Institute of Technology, S J C Institute of Technology, and NITTE Meenakshi Institute of Technology are among the top 10 Aeronautical Engineering colleges in India. Admissions to Aeronautical Engineering colleges in India are based on entrance exams such as TNEA KCET COMEDK UGET etc. Scores. NITTE Meenakshi Institute of Technology and Institute of Aeronautical Engineering are some best Aeronautical Engineering colleges in India as per The Times 2022 Ranking. They are ranked 55 & 64 and offer 120 and 144 B.Tech/ BE Aeronautical Engineering seats respectively. The average fees in the best Aeronautical Engineering colleges in India range from INR 2 - 13 Lacs. Tagore Engineering College and IIST Thiruvananthapuram are the best Aeronautical Engineering colleges in India with low fees of less than INR 2.7 Lacs. And second I will suggest you best colleges for aerospace engineering is -
1. IIT Bombay - Indian Institute of Technology
2. MIT Manipal - Manipal Institute of Technology
3. SRM University Chennai - SRM Institute of Science and Technology
4. RVCE Bangalore - RV College of Engineering
5. IIST Thiruvananthapuram - Indian Institute of Space Science and Technology
6. IIT Madras - Indian Institute of Technology
Do your best buddy and make our Nation and your loved ones proud. God bless you.

Since you have not mentioned the name of your state, it is difficult to accurately answer. However, according to the information given, you may get a Category A seat with a 3000 state rank and EWS category. 516 marks in NEET are decent enough to get you admission to Category A seats in private colleges based on the previous year's cutoff data. Here is an estimated list of colleges where you may get admission for an MBBS course:
1. GITAM Institute of Medical Sciences & Research in Visakhapatnam
2. Nimra Institute of Medical Sciences in Vijayawada
3. The Oxford Medical College Hospital and Research Center in Bangalore
4. Krishna Institute of Medical Sciences in Secunderabad
5. Sri Ramachandra Institute of Higher Education and Research in Chennai.

Yes, you may get admission in NIT Uttarakhand with a 10328 ST rank in JEE Mains, but in the Special Round. The closing rank for the National Institute of Technology, Uttarakhand, in 2021 for most of the courses was under 10,000 for almost every branch of BTech.
Therefore, based on previous year's trends, you may have some chances to get admission in branches like electronics and communication engineering, civil engineering, and mechanical engineering in a special round. The accepted rank for admission in a special round for the BTech in Electronics and Communication Engineering branch was around one lakh.
However, you may not get admission during regular rounds.

There are five states (Rajasthan, Punjab, Haryana, Himachal Pradesh, and Puducherry) where government medical colleges have NRI quotas for NEET candidates. In these states, a few seats, as prescribed by the respective governments, are reserved for NRI quota. Some of these government medical colleges are:
1. Indira Gandhi Medical College and Research Institute, Puducherry
2. Pandit Bhagwat Dayal Sharma University of Health Sciences, Haryana
3. Government Medical College in Kotak, Rajasthan
4. Government Medical College in Patiala
5. Government Medical College in Amritsar
6. Dr. Radhakrishnan Government Medical College in Hamirpur, Himachal Pradesh.