greendesk
The answers to your questions are a bit complicated...
greendesk When I install an app from google play, does the app get anything that identifies me from google, such as my google account email address, full name, address etc?
Unless I'm missing something, you'd have to consent to this kind of thing, most likely by using Google as a single sign on service. If you consent to using your Google account to sign on to a 3rd party website or app, then they can get some data, but Google makes it clear what data you're sharing if you choose to do so.
greendesk for example, does the app developer get a report from google saying something like: jimjones@gmail.com just installed your app!
I don't think so. Google gets the majority of their money by serving ads. Sharing this kind of personal data isn't good for business and is probably illegal in some places. In my experience this kind of thing is usually anonymized, but I don't know about this specific example since I've never published an app to the Google Play Store.
greendesk Likewise, does google get access to any data I might enter into the app? Let's assume it's a fitness app, does google see my workout data?
Not automatically. Google doesn't get special access to an app just because an app was installed via Google Play. In fact, they get no access at all. However, if an app developer uses any kind of Google service, then there's a risk of personal data being shared with Google. An obvious example would be notifications. Say you install WhatsApp. Maybe Google can't read the messages, but they can see how often or when you usually receive them. There are other services that Google very kindly provides to app developers, such as A/B testing, easy crash logs, analytics, ads, and tons of other stuff. It's very easy for Google to get tiny tidbits of info this way.