<<< My first question is: could any of my neighbours pick up on my internet connection? >>>
If you are not actually using the Wireless Network Adapter to connect to a Wireless Network (a wireless router or other wireless PC) then there is nothing for your neighbours to connect to.
<<< Second question: Or does this happen only when someone has two or more PCs in their house sharing a wireless connection? >>>
A wireless network between a single PC and a wireless router could be accessed by "uninvited guests" if it is "unsecured". You need to enable WEP or WPA encryption to prevent unauthorised access.
<<< Third question: If I added a PC and networked them but with a cable, not wirelessly, could a neighbour attempt to use my broadband connection? >>>
No. The only thing a neighbour could connect to is an "unsecured" wireless network, which you don't have. Simply having a Wireless Network Adapter installed in the PC isn't a risk. However, if it would make you feel happier, open the Network Connections folder (Control Panel > Network Connections), right click the "Wireless Network Connection" and select Disable.