I don't think that would work. The wireless router is going to want to send all traffic over the WAN port, and by disconnecting it and connecting it to a LAN port will probably not work - I could be wrong, especially if the router has a "repeater" or "switch" mode. It almost certainly won't work for wireless connections.
Yes, that'll work fine. The router has a WAN port and 4 (maybe 8, but some number) of LAN ports. The LAN ports are switched. So you disable DHCP so the wireless router doesn't send out DHCP offers on the LAN ports, plug the other router into the LAN port so it's DHCP offers get sent out the other DHCP ports, and you're in business. I have that very setup with a Netgear router (DHCP disabled, not using WAN port) and it works perfectly. Wireless included. The wireless radio is switched as well. The routers basically treat it as an additional switched port, except it's wireless, not a physical wire.
The computers connected to the wireless router will want to send out all traffic to the default gateway (the router connected to the modem, 192.168.1.1 probably). So the computers will send their packets to 192.168.1.1. The wireless router (not really routing, but we'll call it that) will see that 192.168.1.1 is off port X (port X is NOT the WAN port, it's a switched port), and will send those out port X, where the actual router will do it's thing, send it back out to port X on the wireless router who will see the destination computer is on port Y and send it there.
The router doesn't even need to have a switched mode. Just statically assign it a LAN IP (I assigned 192.168.1.254 to my Netgear so I can access the management) and disable DHCP and you're in business. A switched mode setting will just do the work for you.
Yes, there are legit reasons for wanting to use 2 actual routers, on 2 separate networks. But for most home users, it's complicated and overkill.
And no need to buy any new equipment when this can be accomplished by changing 2 settings in the wireless router.
Again, this will work perfectly, no additional stuff to buy. I've got one chapter left in my books until I'm ready to take the Cisco CCNA certification, so I'd like to think I know what I'm talking about here
😉