Medicine & Health Sciences Entrance Exam

Hello dear,
With this score, I don't think you may get a government college, because a rank below 15000 is required for a government seat, otherwise it is a bit difficult.

The eligibility depends on the type of course that you wish to pursue, the details of some courses is stated below:
Diploma Class 10th pass from a recognized board with minimum 40% marks.
Bachelor's Degree: Class 10+2 examination with at least 50% marks.
Post-graduation degree: Bachelor's degree from a recognized university with at least 45% marks.
Graduate & PG programs may also require you to appear in entrance exams such as NCHMCT, AIMA UGAT, and E-CHAT etc.

No, you are not eligible to appear for NEET exam.

For general category candidates, it can be around 500.

It depends on your hard work and self confidence that proves that now you are ready for the exam.
According to my point of view, you should be well prepared for NEET, then only give it a try.