I want to know if I can set up my home network like this so every device is on the same network. Basically, the wireless router becomes the router and the DSL Modem/Router does nothing but provide internet:
Is that possible?
Yes. Since your modem also has the router build into it, you would just want to disable the Wireless Router's DHCP server via the web interface. This would allow the wireless connections to pull a local ip address from the DSL Modem's router. THe Wired connections will get an local IP address from the DSL Modem's router.
You would not be able to use your Wireless Router to serve the IP Addresses from its DHCP server while being on the same network as the switch's connections.
Yes, with proper connections and configuration of the DSL Modem/Router and Wireless Router.
That would be possible, but not the way you have it diagrammed.
The connections would be DSL Modem/Router to (WAN port of) Wireless Router to the switch.
If you want to connect the devices exactly as you originally diagrammed it, then you have to use the router that is built-in to the DSL Modem/Router and use the Wireless Router as just as a wireless access point (WAP).
You will have to
This would depend on the router but isnt a very ideal configuration. You should have your router sit between the LAN and the WAN, not as an end-node on the LAN.
Most home routers will have a few LAN ports and one WAN port, and you will probably run into problems using a single interface to have a public and private IP and routing between them. It is possible in theory, but highly impractical and unlikely. If your modem supports routing, you should just disable any routing on the wireless router, and use it strictly as an access point.