Will it be a good idea to write professor who has just interviewed me for a PhD position? I had been interviewed by a professor recently for a Ph.D. position(US). During the interview, he asked me if I had gone through his homepage. Actually I went through superficially as the interview was the very next day after he wrote to me (Also his works are a bit off track with my work). So I answered yes I went through superficially ( I did not want to lie about anything). He seemed quite ok with that and it was kind of a good conversation (maybe he was satisfied with my answers/he was being extra nice-not sure). But before hanging up, he asked if I want I can write to him emails too (he meant further queries).

Is it a good idea to write to him? I mean will it add extra value in my application? (Like I can go through his homepage or publications and ask some questions? or Professors just say it formally?

How long it takes to declare their decisions if I am accepted or rejected? One more thing to share. Some of my scores (GRE, IELTS) is not submitted yet, I mean I sent them but they are on the way). But I got this interview call within two days after uploading my other papers, even before the deadline.
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.