If you have three laptops working wirelessly then you have connections to your router already. You do not need to have a PC connected by ethernet for your other PC's to work wirelessly. If you need to access the router configuration pages for some reason then you will have to connect to the router by ethernet - you simply plug an ethernet cable into a laptop (turn off the wireless) and it should connect as the settings on the wireless laptops are the same as on the ethernet connected pc.
As far as SSID is concerned if you have not made a note of it this can be found on your router's configuration pages.
You haven't given much information on the problem with your old PC - maybe the forum can help with this.