Has Google just stopped supporting Amazon Fire Phone? A few days ago, my phone suddenly started giving me constant "google play services has stopped" pop-ups.

Auto-updated Google Play Services version: 7.3.29 (1866531-036)(7329036)

I tried force quitting, uninstalling, and reinstalling the older version (3.2.25 (761454-36)) along with reinstalling play store app. This made the error pop-ups stop, however, when I tried updating the Google Play Services, it said that google no longer supports your hardware (or something like that). When I tried again, it did update it through play store, then started giving me the constant "google play services has stopped" error again.

I am suspecting that the recent Google Play Services is not supporting Amazon Fire Phone (Fire OS 3.6.8 (128003620)). Is this indeed the case?

Does anybody have an older version of the Google Play Services that work?

Screen captured the message:
This seems to be due to a bug in the current version of Google Play Services, and it's not just limited to Fire Phones. There's a discussion about it over at XDA Developers, which I'd recommend keeping an eye on. 

For now there doesn't seem to be a good solution yet other than uninstalling Google Play Services and reloading an older version - and even then rebooting the phone will automatically update Google Play Services to the broken version. I've seen some commenters say that uninstalling the Play Store will keep Play Services from updating, but I haven't tried that yet.