does game pigeon use data

Being that none of us here are the actual app developers/programmers, no exact way to know. But you can do as security folks do: set up a private Wi-Fi access point just for the phone, turn off cellular data, run only the app for awhile and measure how much data is going in or out the access point's link.

Since it uses iMessage it would use data -- wi-fi if available, otherwise cellular data.

Do you know how much data per text sent or received?

Game pigeon shouldn't use data because texting whether it's iMessage or not. Does not use data