|Why can I not get internet from (Z) using the switch?|
It could easily be you need to use a crossover cable going from the router to the switch. Most modern switches are auto-MDIX capable which means, they'll autodetect the type of cable (crossover as vs a 'straight through' patch cable) and make allowances. But your switch may not be auto-MDIX. Even if it is and that's not the problem, it's standard procedure to always use a crossover when connecting like devices. Essentially, your SOHO router is a switch with either 1 or 4 LAN ports, 1 WAN port, and some services added to it (ie: NAT, DHCP, a firewall) and that's why I say to use a crossover cable between them.
The only thing I noticed was that my patch cables are all in 568B configuration, but that should not matter in a switch right?
Correct, all that matters is that both ends of a patch cable are wired the same way. If your patch cable is A and the cable in the wall is B, it is still 'straight through' from end to end and will work.
To make a crossover cable, you wire one end A and the other end B.
Anyhow, try a crossover cable between your router and switch and see if that doesn't fix the issue. Remember, it has to plug into a LAN port on the SOHO router.