There's no such thing as going to an embassy anymore. Your application will still be processed by a Thai embassy in India, but it will all be done online.
Normally can be used, depending on your immigration office. But there is a distinct risk in using an account like this, as the exchange rate can fluctuate bringing you out of compliance with your visa requirements.
Realistically, what you do in your home online is your own business. Don't go advertising to everyone that you're working and you should be fine. You're far from the only person who would be doing so.
Not for many months. DTV holders are not eligible to open bank accounts as a DTV visa is a tourist class visa, and banks are no longer willing to open accounts for tourists.