Can I install a wireless router and buy a USB card to communicate with the router? - Yes, buy a USB wifi adaptor
Then I can ditch the 10m of cable running from phone socket to PC. - Yes
Also I have the option of a free laptop, again it would need a USB connector as it's not set up for Wi-fi. - Yes
What is the best router to go for? - I've had no problems with Netgear, however some people will slate them. Just ensure that you buy the same manufacturer for the router and the usb dongles (some manufacturers will bundle them together), makes life easier.
The laptop would primarily be used downstairs and 15m and 2 walls away. - Should be okay, the trouble is you will only find out when you have spent out for the kit. See if you can borrow some kit of a mate to test it.
Or am I just talking complete gibberish? No