Q: Can I connect two Airport Express units with ethernet cord and have a third Airport Express receive music wirelesslly via AirPlay? In other words, can I combine a roaming and wireless network?

Using ethernet in roaming network between the two floors is a much better idea than all wireless.

The setup should be automagical if you start with the two AE plugged in by ethernet.

It should be setup in standard roaming with identical wireless names.

If you have issues.. please just post the airport utility screenshots of the setup.. but

One Express is presumably the router??? You need to confirm this.

The second express is then setup in bridge.. That is on the network tab.. off bridge mode.

Wireless is set to create a wireless network and simply use the same name and same password as the other Express.

As stated this should be automagical when you have WAN on the second AE plugged into the LAN on the first AE.

Then wireless extend will also be setup pretty well automatically although sometimes it works better in client mode than extend wireless.

See Client mode.

AirPort Express: How to join an existing Wi-Fi network in client mode - Apple Support

I guess the main thing here is to actually understand what you are doing.

When I tried setting up a partial roaming network before, it was not automagical for me! I wish it had been, but it was not.

The problem with auto setups is that sometimes the magic happens and sometimes not.

To give it the best chance to happen control the environment.. in other words do the setup in the same room using the same computer and hopefully you have a few spare ethernet cables and a computer with ethernet port or adapter.

Q1: Before setting up the new network, should I dismantle the current, wireless network? In other words, should I used AirPort Utility to tell my network to forget all of the additional AEs other than the Main AE Base Station?

Don't worry about the airport utility.. it will dynamically adjust to the setup. Reset the additional AE.. I always hardware reset, ie use the reset button which needs a paper clip bent out or a toothpick etc. You can also reset via the AU.. merely forgetting is not good enough.

Q2: How do I confirm that one AE is the router? This will be the Main Airport Express Base Station, which is connected to the cable model via ethernet

In the airport utility click the AE. This is the short summary screen. There is also a full summary screen by holding option key and double clicking the icon.

If the AE is in the right mode you will see two different IP addresses.. a wan and a lan IP.

See the pic below.. this is bridge mode.. it has the same IP on wan and lan hence the repeated numbers.

Click the Edit button.. marked clearly for you.. and go to the Network tab.. that will show you exactly the configuration of the AE.

For your connection with a pure cable modem it would be DHCP and NAT. Don't worry about my numbers.. yours will be different.. but the Router Mode should be as shown.

Q3: While setting up the Main AE Base Station, under the Internet tab, I am offered 3 options for Connect Using: (a) DHCP, (b) Static or (c) PPPoE - which is correct?

For a standard cable modem it should be set to DHCP.

If you have a modem router it will still be DHCP.. but you have another problem if that is the case.. and the Network tab needs to be set correctly to bridge.

Q4: Again, for the router, or Main AE Base Station, under the Wireless tab and Network Mode, I am offered 4 options: (a) Create a wireless network, (b) Join a wireless network, (c) Extend a wireless network, or (d) Off - which of those is the correct choice?

Always create a wireless network if you connect by ethernet.. Extend a wireless network when you connect by wireless. In the later case no ethernet can be connected.

Q5: Under the Network Tab, I am offered 2 options: (a) Off (Bridge mode) or (b) DHCP and NAT - which of those should I choose, keeping in mind that this is for the router, the Main AE Base Station?

As per the answer to Q2.. DHCP and NAT if the modem is standard single ethernet type. If the cable modem has 4 ethernet ports and wireless it should be bridge.

Q6: what setting will be different for the 2nd Floor Base Station, which will be connected to the Main AE Base Station via ethernet cable?

The AE2 will have bridged on the Network tab.

It will have create a wireless network using the same name (SSID) and same password as the main AE.

Any issues give me screenshots.. we can easily correct it.

Glad you got it working.. and it is good that opposites can help.. my Maths is hopeless. Though I did Engineering at uni.. 40years ago.. I assure you what I can do on a basic calculator is it.

Glad to also hear you got it working to your satisfaction.

Guest network is a strange aberration in Apple routers.. all the other brands I can think of provide guest wireless in bridge mode.. apple cannot. It works poorly anyway so I don't think it is a huge loss.

Why not use the wireless in the cable modem router for guests? Many people are over worried about using separate wireless networks.. but that is fine. There are sufficient channels to operate 3 wireless networks in close proximity to each other.

If you have turned off the wireless in your cable modem turn it back on.. and check the manual because it could also have guest functionality.. regardless it can still be used as a guest network whichever way you do it .. just look at the security of your network if there are guests who not trustworthy. The proper guest functionality removes the ability of the guest client to access anything but internet. However your Apple systems are pretty difficult to access from a PC unless you have turned on permissions..

Even if that doesn't work you can just buy a cheap wireless router and plug it into the main router as guest wireless.. you can buy those units for $40 or less.