Yep, lots of different options depending on exactly how you want to set things up and how much you want to spend. Getting the ethernet in between the new wi-fi and your existing router is the important part.
Great. Good to know. I'm not naïve enough to think I can just plug something into the socket and expect it to appear as a network that I sign into with my existing WiFi password, but that would be the dream.
Yep, lots of different options depending on exactly how you want to set things up and how much you want to spend. Getting the ethernet in between the new wi-fi and your existing router is the important part.