Do we need any kind of permission to add a room in a house in USA?

I guess you need to have a permit to do any modification to your house if you want to add it legally. If you are living in a HOA managed community then any change in exterior needs their permit as well.

Thanks Stylish. Its not under HOA community. And the extra room is just like a partition of free space I have available. Still I need to get a permission?

well, it depends upon city you live in. Areas in US are monitored by different jurisdictions. You could check with public works department of your city.

step 1 : Talk to your HOA (they give you an idea ) if exists. step 2: almost all city has Public work department usually they regulate this things. If you live in california, new york states they have very stringent requirements to get permit so might be very expensive but in some states eg arizona, texas might be super easy. You can talk to local contractors too so they guide you because the have hands on experience on that.

NO, if you live outside the city limits. YES, for everything if you live inside the city limits.

Depend on county ordinance. You should go to county office and inquire about it