Thanks for that explanation. I guess I don't really know the right questions to ask. I assumed since each device is set with a DNS entry, and that seems to always be the gateway's address, not the ISP DNS, that the DNS setting determined which device directs all the traffic.
By default, each network device is a DHCP client. As such, it gets its IP address information, including where to find the default router and DNS servers, from a DHCP server. That server, is most likely, your ISP-provided gateway device ... thus why each device is "told" that the gateway is to where to look for DNS service. In turn, your gateway is getting its IP addresses from your ISP ... as it too, most likely, is configured as a DHCP client.
Hopefully, I didn't add to the confusion there.
That's my real question then: if I transfer data between two devices that are connected to a router that's downstream from the gateway, does that traffic go all the way to the gateway and back to the router, based on some setting (DHCP? "Router" address in network settings? NAT? something else?), and can I change that setting so that the router in my kitchen is the traffic cop rather than the gateway?
Again, for any devices on your local network "behind that gateway," the gateway's built-in Ethernet switch is doing all the work to move data between these devices. Yes, the data it going "through" the router, but only the switch portion. Neither DHCP nor NAT comes into play. Each network device has a pre-defined MAC (hardware) address. The switch "remembers" these addresses and to which port on the switch they are connected to. IP addresses are not used at all in this case, just the MAC addresses are required to move data between devices. Staying awake? Sorry, I know this can get really boring quickly.
Only when a device needs to access another device that is NOT on the same local network, do IP addresses matter ... and this is where the router portion of the gateway has the work cut out for it. The router, similar to the switch, keeps track of a device that is sending data through it, but instead of keeping track of MAC addresses, it keeps track of IP addresses.
Where DNS comes into the scene, is to think of this service as the Internet's version of the Yellow Pages. DNS keeps track of a site's URL (aka, www.apple.com) and its actual IP address (XXX.XXX.XXX.XXX). URLs are intended for humans as computers only understand the equivalent IP address, otherwise, you would need to use the site's IP address to access it.
So where DNS does have an effect is getting to Internet web sites. Some DNS servers do a better job at getting you there quickly ... and your ISP-provided ones may not be the best available for you to use for this. That's why you can try the Google DNS or OpenDNS servers to see if they help improve your access to Internet sites.