B.A

To take admission in the University of Mumbai, you will have to fill out the form available on the university website and on the college website that you are aiming at and then they might conduct an interview.

You have to clear NEET and it also depends upon the category to which you belong.

Yes you can get admissions in some of the colleges and by the way it also depends upon the category in which you are in.
Some of the colleges of BHMS and BAMS are:
1. Uttranchal Ayurvedic College.
2. Uttrakhand Ayurveda College
3. Beehive college of advanced studies.
4. Doon ayurvedic.
5. Shivalik institute.

It will be quite hard for you to take admission in SD College for BA (Hons).

If you want to become an IAS officer then I think that doing graduation in BA would be a better option and it would be truly appreciable if you could take political science or history.

For a guaranteed job in India, most important is communication or you can say English and which can secure a successful career is, undoubtedly, maths.
Good wishes.

IGNOU and SOL are two universities which offer distance/ correspondence study in various courses accept BA or B.Com.

For better career prospects after BA, you can opt for a Master's degree programme. M.Sc (Traffic Management) is one of the most popular options as far as a Master's degree is concerned. The eligibility for an M.Sc (Traffic Management) is graduation in any field. This course covers various domains such as:
1. Traffic Engineering
2. Forensic Science in Crash Investigation
3. Human Factors in Road Crashes
4. Driver Training and Management
5. Post-crash Management
6. Traffic Legislation and Codes of Practice
7. Traffic Enforcement 8. Road Safety Awareness and Education

First of all, select the university and program you want to apply to. Next, check the admission dates & deadlines. Now, complete the online application & pay the registration fee. You will get intimation for the interview.
Normally, it's easy to get through the interview and you get admission.