I am an Indian and I am now working as an online tutor in a us company as an independent contractor.For the payment transactions they put forward two options - check by mail and direct deposit. Which one you prefer for me? If I select direct deposit,What are the requirements( bank matters) I need to take from my side? Whether I need to pay any tax to US government or Indian government or to both? If yes what is the rate? Which all banks they may prefer the most for direct deposit?

If you are not a resident of US and you were not present in US to work, the income is not taxable in US. It is taxable in India. The US company may require you to file Form W-8BEN.
Your other queries are not tax related.

We don't care how you get your money, a written check takes more time to be mailed. Direct deposit is normally there the same day. Most banks all work for direct deposit, you will merely supply them with all the tracking numbers and account numbers.

The ONLY drawback for direct deposit is that the payor can reach in and direct debit the account if they determine you were paid in error.

Hi
I am in the same condition, working for a US company from India. I had filed W8-Ben with them. Should they be issuing me any document such as 1099-MISC or not. Or should I just ahead and file my taxes here with that income.
Thanks

You should NOT be getting the Form 1099-MISC.

File the income on your Indian return.