BHMS

Yes, it seems to be quite possible as there are total 26 medical colleges for MBBS considering 250 of them have average seats then there is still BHMS, BAMS, etc. last year cut off of BHMS was approx. 13500 so you are eligible

Homeopathy is a great treatment method and is the 2nd most used treatment method. It has great scope in future. To begin with, you need to work on the entrance exam, NEET. Then opt BHMS degree from a reputed medical college. Then after degree, you need to pursue MD.

No, you can't get lateral entry admission to MBBS after completing BHMS courses. Both the courses are different from each other though they belongs to same category.
Also note that there is no lateral entry in MBBS.

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.

There are ample of job opportunities in BAMS. So, go for it.
Good luck.

You can apply for UPSC when you are in the last year of your course. You can apply next year for UPSC. You have to make sure that before interview you must have all the mark-sheet and degree. The candidates who have their results awaited can also apply. You can apply next year for UPSC.