Do iPhone all apps connect to the internet?

I've downloaded a couple of games, so I can play when I'm not connected to the Wi-Fi. I can't imagine games would need to connect to the net but just in case, is there somewhere in the app description that will tell me if it uses the internet.

With apps like facebook, twitter, and skype apps, do they only access the internet when I open them or all the time so they can notify me when I'm not using them?

No not all

Not really like this, it will use internet only when you are playing theses games otherwise it wouldn't use internet services.
So feel free.

Not all games use the internet. If you are playing single player events, games will not require the internet. But if you are planning to play online multiplayer or get score updates from the Game Center or other online community, you will require an internet connection.