Printing
In general, any printer in a standard network in not going to need or use the difference between Gigabit Ethernet vs standard 10/100. Unless this is going to be a massive print server hit by lots of people all the time, don't worry about it.
Anybody who is using wireless (a/b/g/n) won't be approaching the Gigabit speed anyway, so it is only an issue at all for the wired folks.
IP Addressing
Using a standard switch or Hub behind the router is fine for DHCP. The RT-N66U (or any other standard DHCP server) will serve up addresses to the subsidiary switch's connections.
The only limitation is if the switch has one port that it wants to use as Uplink (or some similar term). If it has a special port for the Uplink, plug an Ethernet cable from one of the RT-N66U's LAN ports to the Uplink port on the switch, if it exists. I checked the docs on the DS-108. You have to connect the router to Port 8 on the hub and set the Normal/Uplink button to the Uplink position.
If it doesn't have a special port for Uplink, plug an Ethernet cable from one of the RT-N66U's LAN ports to any port on the switch. My preference has always been to use the highest number port on the switch, leaving all the lower number ports for workstations and printers.
Switch vs Hub
Your Netgear DS108 is ancient (circa 2000). It is a 10/100 hub vs a 10/100 switch. I checked. It will work fine for connecting the printer and a couple of light-duty workstations. The main difference between a switch and a hub is what happens when there is a lot of traffic from different devices. The more traffic going to and from different devices, the more the throughput decreases to all devices.
My suggestion would be to spend the $50 or so and get a new switch (or at least a newer one from eBay or something.)