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IANAL, but my understanding is HIPAA applies to covered entities, i.e. medical professionals.

Google can use the defense that somewhere in the TOS it probably says you agree to share everything with them when it goes to cloud storage.

HIPAA doesn't prohibit people from sharing info about themselves (even accidentally). Though it does probibits your doctor from putting photos like this near Google anything.

HIPAA specifies how medical professionals should handle patient information. None of the parties in this dispute are medical professionals, so HIPAA does not apply.

No. Google Photos under a personal Google account is not HIPAA compliant and doesn't claim to be.

I guarantee that the doctor has a HIPAA agreement that you sign when you agree to communicate over email that says that you grant him the right to share the files with Business Associates, and that their email providers are considered Business Associates for the purpose of transmitting ePHI.