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Yes you can. The AEEE ranks below 10000 under state quota are very good enough for getting Coimbatore, Amritapuri and Bangalore campuses.

Your JEE Mains rank is very good enough for getting desired engineering branch in Amrita Vishwa Vidyapeetham for Coimbatore, Bangalore and Amritapuri campuses.

Please write about the home state you belong to. Based on your JEE Main percentile, following colleges can be the good options for you:
1. Thapar Institute of Engineering and Technology, Patiala
2. Chandigarh College of Engineering and Technology (Mechanical)
3. Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal (Automobile)
4. Barkatullah University Institute of Technology, Bhopal ( ECE, IT& Mechanical)
5. Govind Ballabh Pant University of Agriculture and Technology (CSE)
6. I. K Gujral Punjab Technical University (PTU), Jalandhar (ECE)

Yes, it is legally valid and you can choose different options regarding that.

Yes you can take admission in many private colleges.

The MS abbreviation is not enough for providing the correct information. However Armed Forces Medical College (AFMC) is a medical college in Pune, India, in the state of Maharashtra. The college is managed by the Indian Armed Forces. The AFMC Pune is offering various UG and PG level courses like MBBS MD/MS, Super speciality, Paramedical courses and Nursing courses etc.

Use this college predictor tool
https://www.shiksha.com/engineering/jee-main-college-predictor?sg[]=113&sg[]=111&fromWhere=form2&origin=form1
I think if you are not a dropper you should take a drop.