proof of concept demo with iron-form
This is a first pass at demonstrating putting a required google-recaptcha element inside an iron-form. To accomplish this, I gave the google-recaptcha element the Polymer.IronFormElementBehavior and Polymer.IronValidatableBehavior behaviors and supporting logic.
This may more properly belong in a separate element just for iron-forms (e.g. iron-form-recaptcha) if you'd like to keep google-recaptcha agnostic to the form implementation it's included in, but it seems there should be a generic way for the google-recaptcha element to advertise its validity state to any containing form implementations. Polymer developers please advise, and I will be happy to help with taking this from proof of concept to whatever you envision offering.
What about set pristine to false in _responseHandler and _expiredHandler. It can make sense to have visual rendering even if you havn't click on the submit button.
I think you should remove class="foo" from template tag :)