I found the solution to fix the Google account problem.
Here's what I did:
1 - I launched the unmodified game and pressed “clear cache.
2 - Still on the non-mod application, I logged in via my Google account and it worked.
3 - I uninstalled the non-mod version.
4 - I installed the mod version and logged in with my Google account, and it works.
I think it's possible to clear the cache directly from the mod version (haven't tried it), but if it works, it removes a few steps
Sorry for bad english.