Class 11

If you have your mark sheet of 12th that will be considered for admission purpose.
11th marks card is mandatory only if you don't have your 12th marks yet. Just keep it with you though, in case they ask for it.

You should start preparing in class 12th.

You should consult with your school authorities whether you can switch your stream to one with biology. I am sure if you are good in studies and have a proven academic track record, you should be able to get approval to make the change.

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Yes, you can start coaching in maths. You can also go for M.Sc in maths as you have interest in maths. You can also try for MBA and then try for startup in education sector.

TOEFL- If you are proficient in English, then you should not spend more than a total of 15 hrs. Just solve few sample tests. Same goes for IELTS. If you are moderately proficient, spend 2 hours every day for 3 weeks.
For SAT and ACT: It depends on your proficiency and your target score. If you are aiming for the top 10 percentile and also are quite proficient already, then spend 3 hours for 2 weeks solving papers and revision. If you have just started preparing, then spend 2 hours for 3 months to gain good understanding of the test pattern and to do well.

The choice is entirely yours. If you feel you do not want to pursue science anymore, you should pick a stream which you would like to study. You should also consider if you would have to repeat a year, or if you would get an immediate transfer to a stream of your choice. If your grades are good, you most probably won't have to drop a year. Consult with your teachers whether you would be able to cope in another stream.

Rather than pursuing any extra courses, you should actually focus on the preparation of the 11th and 12th.Or in case, you aspire to appear for any entrance exam, then prepare for that. This will be the foundation of your career, so the results will matter. You will have a lot of time to pursue additional courses ahead.