I believe Teamviewer connections are shoveled through the Teamviewer servers. Otherwise, I wouldn't know how most client edge devices would know to route to a client using PAT/NAT.
But yeah, like the other answer stated, it would be straightforward to check the open connections yourself to inspect the IP handle.
As for your other questions.
I'm not sure what you mean by this one. Are you talking about referring website and general tracking information? I would suppose your best bet would be to browse in Incognito/Private mode or any of those specialty anti-tracking browsers.
I wouldn't see how, but I suppose it is possible the teamviewer client could leak/expose this information, though I'm not sure what it'll get you outside of the client version. All in all, I wouldn't count on that information to be confidential.
MAC Addresses change as you hit each switch, Those MAC tables are how the switches forward the packets. A cursory look, this may help you understand.