I have Verizon Fios
Then, you have a modem/router or gateway type of device, not a simple modem.
My switch is the Cisco SR2016.
Thanks, this confirms that you have an unmanaged switch......meaning that it requires no programming.
My Base AirPort Extreme was just purchased so I believe it's ME918LL/A.
Thanks, got that.
My Extended stations are various older models of AirPort Extremes: one tower and one flat (the tower doesn't state the model)
Thanks, got that.
Can the Extended Base Station be plugged into a switch (that the Primary also plugs into) instead of the Primary Base Station directly?
Yes, assuming that all of the AirPorts are set up to operate in Bridge Mode. So for now, power off the device that you call the "extended" station and check to make sure that all the AirPorts are set up to operate in Bridge Mode, which Apple calls "Off (Bridge Mode)".
We asked for a layout of your network, but did not get an answer to that question. For now, we know that you have a Verizon modem/router. That is the next device connected to the Verizion modem/router, and what connects to what after that. For example, do things look like this:
Modem/Router > Main AirPort > Switch > Other AirPorts
or
Modem/Router > Switch> Other AirPorts all connect to the switch
In other words, what connects to what on your network?
We assume that you have a Mac using the current or recent operating system. Is that correct?