The answer assumes that your runnig the the tor deamon on the same host as the browser and that DNS lookup are done via Tor. Both holds true if you use the Tor Browser.
If I visit http://expyuzz4wqqyqhjn.onion/ through Tor Browser, does my router see that address or does it only see the first relay Tor Browser connects to?
It only sees the address of the first relay and it sees the traffic that you exchange. It can't read the traffic however, it is encrypted.
Does it see anything different when visiting a non-hidden service address like google.com?
No, the address of the host you're visiting is transferred encrypted.
And is there a difference in what address it sees with/without https
Ditto. However, keep in mind that when you use http to connect to a clear net host, the traffic can be read and manipulated by the exit node and anyone between the exit node and the destination host.
I've read similar questions here but the answers were all relating to what the ISP can see, not your router. I assume the two see something similar if not the same, but I want to be absolutely certain.
There are a few pieces of information that your router may see your ISP doesn't:
- your MAC address (which may uniquely identify the device you're using)
- LAN traffic (In particular, it may see broadcast traffic coming from your host which can contain identifying information like the host name or OS.)
- DHCP request (can contain host name and your IP address from when you last connected to the network)