MBBS

Yes, you can get admission in either MBBS or BDS with a 2.82 lakh all-India rank in NEET. However, the NEET admission cutoffs vary every year in Maharashtra depending on various factors, like the number of seats available and a quota-based closing rank. Therefore, I am giving you a list of estimated colleges where you might get admission with your rank in Maharashtra. According to the previous year's closing ranks at different medical and dental colleges, you can get admission into the following colleges:
For the MBBS course:
1. Dr. D. Y. Patil Medical College, Hospital, and Research Centre, Pune
2. Padmashree Dr. D. Y. Patil Vidyapeeth, Navi Mumbai
For the Bachelor of Dental Surgery (BDS) Course:
1. Dental College and Hospital (Bharati Vidyapeeth), Pune
2. Navi Mumbai's BVDU Dental College and Hospital
Also, your admission is subject to other criteria as set by the college, such as minimum qualifying marks in the 12th.

Yes, with a score of 565, you may get admission into the MBBS course at some very good colleges, including government colleges. It has been observed that for the OBC category, there are some chances to get into a government medical college for MBBS even if the usual cutoff is around 600 marks. For instance, these are some of the medical colleges where you might get admission based on a score around 565 on the NEET-UG:
1. Government Medical College, Akola
2. Guwahati Medical College, Guwahati
3. Rajiv Gandhi Institute of Medical Sciences, Ongole
4. Data Meghe Institute of Medical Sciences, Wardha
5. MGM Medical College, Jamshedpur
This estimate is based on the 2021 cutoffs, and this year's actual admission may be affected by the number of students applying in your category and seats available.

First of all, congratulations!
Yes, you can get a seat very easily at the Institute of Post Graduate Medical Education and Research (SSKM) in Kolkata.
85 percent of seats are reserved for state quota students. SSKM College opening rank Under state quota, the expected score is 545 and the expected cut-off is 21,000 for the EWS category. SSKM College closing rank Under state quota, the expected score is 570 and the expected cut-off is 21,500 for the EWS category. The Institute of Post Graduate Medical Education and Research (IPGMER or SSKM) is located in Kolkata, West Bengal, and is affiliated with the West Bengal University of Health Sciences, Kolkata, and approved by the National Medical Commission (NMC). It was established in 1957.