Class 11

Yes, you can pursue this course as this requires 10+2 pass from a recognised board.
To get the details about the course, refer to the link: http://isasbeautyschool.com/post-graduate-diploma-in-cosmetology/.
The course structure will give you an idea what exactly is taught in this course.

It may vary from 50000 to 1 lakhs but don't worry many coaching institute provides scholarship discounts of 10% to 75% depending on your percentage of class 10th and entrance exam conducted by the institute.

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.

Hi,
Kindly visit the link mentioned below:
https://www.shiksha.com/career-in-commercial-flying-all-you-need-to-know-article-13472-1

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.