Question – I am going to US on a f visa can

I am going to US on a f24 visa, can i work for an indian company remotely and get paid in my account in india while i wait for my greencard and other related paperwork once i enter US

Question – If an F- student pursuing a Ph D degree is

1. If an F-1 student pursuing a Ph.D degree is unable to complete his study by the program end date on the Form I-20, and is granted an extension by the DSO, must he apply a new F-1 visa ? Whether he can continue to stay till finish his study without applying a new F-1 visa ?

2. If he will do post-doc research after his Ph.D program, is his status F-1 or F-1 OPT ? What is the best way for him to maintain his status in the US ?