I would like my Actions on Google agent to store and retrieve certain pieces of information across invocations - like a cookie. How do I do this?
Sounds like you may want to checkout out Account Linking: https://developers.google.com/actions/identity/account-linking. With account linking you can collect end-user information which you exchange with Google by providing a unique key. This unique key becomes part of every request you receive from Google, so when you get that unique key you lookup the information you collected from the end-user. In your case, you would store credentials or whatever key is required to access the end-user information. After the initial linking, any new data you obtain could be stored along with the original information collected, based on the unique key obtained during account linking.