Your problem with Achievements is likely related to having the incorrect signature for your APK in the API console. If the signature is wrong the sign in dialog will appear, and then immediately disappear without logging you in. This is a "security feature" of the Google Auth API.
If you created your keystore using the tool in Construct then we would have showed you the signature of the keystore at the time. However, if you are using "Google app signing" then the signature will change when you upload the APK to the play store.
You don't have to publish your APK to the store to test that this works, but the APK must be signed with the keystore that you specified when setting up G Play Games.
The other issue sounds related to this. We have already updated to the new newer billing library in Mobile IAP.