Science

Yes, you can do M.Sc in mathematics through distance education.

You have to go to the respective website where you can get all the information regarding environment management courses, entrance exam and syllabus.

Yes, you should be expecting a regular seat with this rank because you have secured a good rank.

No, it is not mandatory to do the integrated program but you have to apply for the M.Sc. You just apply for it and don't go for the integrated courses.

According to your score in BHU maths group and according to my knowledge, you will get a regular seat in any of the affiliated colleges.

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.

Can you please elaborate your question because I am unable to understand what you have asked?

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.

You can use the below tool to clear your doubts:
https://www.shiksha.com/college-predictor

You might find it a bit difficult in the initial days, but over the duration of the course, you would be able to understand just like any other biology student would. But you would have to put some effort. It's up to you to decide whether it's worth putting the extra effort.