Considering this information there does appear to be a problem with your network but that doesn't mean it's a hardware issue (although don't rule that out).
Firstly the concern about your neighbour's network. This is not going to cause issues because you are not connected to it as such there is no way the conflict would be caused by your neighbour. Just because you can see it is immaterial so you can rule that out. The only way his network could cause problems is if his network had a stronger signal than yours but that seems highly unlikely.
Here is a document I drew up that outlines the procedures for ensuring the machines are set to DHCP. It's imperative that no machines have an IP address that is the same. It's preferable to not have any machines with a static IP Address because if it isn't set correctly it can cause an issue where the machines set on DHCP are being served the same address which I suspect may have been the problem from the outset.
TCP/IP Settings
Also, can you provide me with screenshots of the configuration of each of the screens on your Airport? Send them to:
lowededwookie at mac dot com
That way I can see what's going on.
Favoured: 0
