Right, let's get some terminology straight

The router is now acting as a switch (since it is not routing the internet any longer).
The DHCP server can reside anywhere but it must have a fixed ip address outside the pool of dhcp addresses it is going to give out. The DHCP server does not have a name.
Access points are closer to being hubs than switches and are definitely not routers!!
The gateway is the IP address of the machine (be it nic in a pc or a router).
All names should be in UPPPER case and contain no spaces or special characters
All names should be different for each piece of equipment on your own network so that you can use the \\server\sharedDevice convention

Keep it simple
Gateway PC - call it what you will - I would call it IGATEWAY
Access Points have SSIDs - this is NOT the same as a UNC Name (\\server\share)
Routers don't have UNC names

192.168.0.1 by convention is the gateway NIC
On your gateway machine, you will have two ips - one for each nic - what's the other nic getting for an IP?
Order your network by number so that IP addresses can be easily identified.
So the router which is being a switch I would put at 192.168.0.2
I would then leave a gap and start the Access points at 192.168.0.5
I would set the dhcp pool from 192.168.0.20 - 50 so I had plenty of addresses

On the client machines, nothing should be set - everything should be done via dhcp
Make sure that IE or whatever browser they are using doesn't have any proxy settings and untick the "automatically discover settings" in the browser.

How is the internet connected to the gateway pc? Usb modem? Ethernet Modem or something else?