BAMS

Hi Arbaz,
You may not find a typical BAMS program but you can find alternate medicine programs. Ayurveda programs are getting popular in many countries including US but still you will find options as in India. Check American University of Complementary Medicine,Ayurveda Institute of America,Florida Vedic College, The Health Institute Australia etc. Please visit the websites for programs and other details.

In NEET 2016, the marks range for unreserved category under All India Quota seats was 685 to 468.

For BAMS course you need to appear for different national and state level entrance examination like KLEU, AIET, KLE University,
Andhra Pradesh: EAMCET, Andhra Pradesh Engineering, Agriculture and Medical Common Entrance Test.
Assam: CEE, Assam Combined Entrance Examination,
TS EAMCET: Telangana Engineering, Agriculture and Medical Common Entrance Test.

You need to appear for the medical entrance examination. You need to have PCM stream with an aggregate of 50%. The other procedures are similar to that of medical entrance examination.
The colleges that offer BAMS select candidates from the Medical entrance rank list. The duration of this course is 5 years.
All the best.

No, you can't get admission for BHMS, BAMS without competitive exams.

As per Hon Supreme Court of India, all medical admissions will be done through NEET merit list whether MBBS, BHMS, BAMS or BUMS.

The scores of NEET exam varies from institute to institute. You will have to check with the institute in which you want to take admission.