Is there a way to see/copy the email of the person I am talking to via an opened little Hangouts chat in the Gmail web client?
I don't know of any way to see the email in the chat box. But in Gmail on the side you have a little contact list of people you have chatted with.
You can hover over the name and see the email.
There is no way you can access/copy the email from the hangout window. Even from the contact list you have to hover on the contact to see contact's details. These details are fetched on the fly.
Technically, the chat window is inside an iframe. You cannot access iframe content.