Last question first.
Can I connect the base unit directly to the internet service instead of using the Pace modem, or does it HAVE to be connected to the Pace?
No, because the Pace device is a modem/router or gateway device that is actually two devices.....a modem....and....a WiFi router..... combined into a single physical package. You might be able to turn off the wireless function on the Pace modem/router though, if you do not want to use the WiFi signal that the device provides. Or, you might want to reserve the Pace wireless for "guests" to use.
The AirPort Extreme is a router only......there is no modem on board.....so while you might be able to physically connect the AirPort Extreme directly to your Internet wall port, the AirPort Extreme would not be able to connect to the Internet.
However, when I connect the base unit (by connecting an ethernet cable from a Pace port to the WAN port of the Extreme) and the unit displays a green light (indicating it is connected properly and working), I do NOT see the base unit as a network to choose from in Settings/WiFi on my iMac,
If the AirPort Extreme was set up correctly, you entered a name for the wireless network for the AirPort Extreme to broadcast along with a password for the network. Do you remember the name of the network and the password that you assigned for the network?
If you do not see the name of this network when you click on the WiFi menu at the top of the Macs screen, then one or more of the following possibilities is likely true:
1) The Airport Extreme was not set up correctly.
2) The AirPort Extreme wireless radio is not working, or it has been disabled in the settings for the device.
3) Unlikely, but if you enabled the option on the AirPort Extreme to "hide" the network name, then the network will not appear when you click on the WiFi menu at the top of your Mac's screen.
Does your Mac have an Ethernet port that you could use to connect the Mac to one of the three LAN <--> Ethernet ports on the AirPort Extreme.......to check some settings that way?
If not, a Thunderbolt to Ethernet adapter is a mighty handy item to keep in your desk drawer.....so you can check things using a wired connection when wireless is acting up, or not working.
Thunderbolt to Gigabit Ethernet Adapter - Apple