Ranges for Multiple DHCP Server: Better to overlap or separate? wireless vs ethernet clients?
Solution 1:
A DHCP server will service all the requests it receives. If the wireless Ethernet network and the wired Ethernet network are the same broadcast domain then you don't want to be running two DHCP servers in that single broadcast domain (in the vast majority of cases). If they're separate broadcast domains (implying that they're different IP subnets, too) you can run a DHCP server on a host in each broadcast domain, or if the router separating the subnets supports DHCP relaying you could run a single DHCP server and receive relayed requests from the clients on the "far" subnet.
I suspect you've got a single broadcast domain, since you're talking about "overlapping" the address scopes. That being the case you really should have only one DHCP server servicing that entire broadcast domain. (There are situations where you could have more than one in a broadcast domain-- typically in a failover scenario-- but that's beyond the scope of your question.)
Solution 2:
As Evan mentioned you don't want two DHCP servers on the same broadcast domain. Which DHCP server responds to a client's DHCPREQUEST/DHCPRENEW is whichever one gets the DHCPREQUEST first. Neither DHCP server is going to be aware of the other's lease database (which clients have which addresses) so you're almost guaranteed to get IP address conflicts and just like Highlander there can only be one! (Things won't work right if you have the same IP address assigned to more than one station).
The exception to this is doing DHCP Split Scope. The idea is that half of your available addresses goes to one DHCP server and the other half to other. This way you can guarantee that you won't end up with a the same address assigned to multiple clients AND you get some redundancy in your DHCP services.
Also, would it be possible (and advantageous) to have my wireless router service only the wireless requests, while having the Ubuntu Server Box service the Ethernet ones?
If you had more than one broadcast domain this can be nice. You can subnet your addressing space and give one subnet to the wireless clients and another to your wired clients. This makes routing, filtering, and monitoring them easier.
EDIT: I just looked at your other question. What you want is either a separate network (either physical or VLANed) for your FOG clients with one subnet and associated DHCP services and another network for your employee machines (with their own services).
Solution 3:
What about setting the router in the Access Point mode? DHCP responses will be passed from the Ubuntu Server to the wireless clients directly.
Basically every wireless router supports the AP mode, just disable DHCP server on the router, enable WiFi radio and plug in the router to the network using one of the LAN ports.