How do I convert an existing WiFi network to a wired ethernet connection? I'm not just looking for a single connection (i.e. WiFi bridge). The WiFi is coming from a mobile device (internet sharing) with a limited number of clients. It would have to be something like a "bridge-router" combination, with a DHCP server that is independent from the mobile device. Does such a device exist?
Yes, such devices exist: basically every home router that has both ethernet ports and wifi can do that. You can get used ones for very cheap, and even new ones are not expensive. It doesn't need to have DSL, but having it doesn't hurt.
Put OpenWRT on it, then you can configure it with a DHCP server and NAT, if the original firmware doesn't allow you to do so.