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Johnny I have both the Aurora Store and Google Play Store installed in addition to sandboxed Google Play Services.
- Is there any privacy advantage to downloading apps from the Aurora Store?
The advantage of using Aurora Store as an anonymous front end to access the Play Store is that you can install sandboxed play services, and initialize them per the instructions on the site without logging in, all while installing apps that make use of them all without a Google account. This comes in handy should you not be able to create a throwaway account as Google asks for your phone number on sign up or you don't want to pair it with a personal account to use the Play Store.
- If I download an app from the Google Play Store, will it have access to my Google account information?
No, apps only have access to information you provide them, for them to have access to your Google account info you would have needed to use some part of it to register for a service or use the Google SSO (Single Sign-On) button to login to the app/service. Play Services will have access to your Google account info if you used a personally identifiable account to login to the Play Store. However Google will only know how you use their apps. The only information as I understand it that Google gets from apps using Play Services is each apps unique session ID and your IP to know where to route push notifications etc.