When you plugged the WAP into the switch, did you get a link light on both ends? If not, it could be a cable problem, maybe you need a cross-over type.
Is the WAP giving out IP addresses within th erange of your home subnet? Does the DHCP range overlap with any devices that have a static IP address?
The other thing with DHCP is that you also have to correctly define the DNS server and gateway that the DHCP clients should use. It usually defaults to the WAP itself. You may want to change that to the IP address of the Linux NAT box.
You could also turn off DHCP on the WAP and make the Linux box do it for you.