You don't need cell service to install games. If you are trying to install Apps from iTunes on a computer, make sure the versions of the Apps on iTunes are ones supported by the iPhone. Some games may require iOS 8 which your iPhone 4 cannot support.
Once you press the install button on the apps, you need to click on the Sync button at the bottom to initiate the install. The install/remove buttons just set markers on the Apps for what action to perform on the next sync. They don't instantly install or remove the Apps.
Alternatively, connect the iPhone to Wifi, and go to the App Store App on the iPhone, tap on Updates, then Purchased, and look for the games there.
This will let you download compatible versions of all the Apps directly onto the iPhone.