Will it be a good idea to write professor who has just interviewed me for a PhD position?
If you have questions then ask them. Otherwise a simple Thank You note would be appropriate. But just asking questions in an attempt to pad your application probably won't do any good. But thank him now and ask questions later if they arise. 

How long it takes is variable and up to them. But they need all required information before they make any decisions. There is probably a committee that has to get involved in such decisions in the US, also. 

Patience. Keep all your other options open.