Use Time Capsule (AirPort Extreme) to extend wifi coverage of Xfinity X1 modem/router

Do the cables all need to connect directly to the Time Capsule itself

That would be ideal, but you can "daisy" chain devices together if that is easier. The downside to doing this though......Say you have the Flat AirPort connected to the Time Capsule and the other two AirPorts connected to Flat. If Flat has a problem, then so too will the other two AirPorts have a problem......and it will be difficult to try to figure out where the real issue might be.

If all of the AirPorts connect directly to the Time Capsule and one of them has a problem, the other AirPorts are not affected, so it makes troubleshooting much easier.

I am concerned about having enough ports on the Time Capsule

You can add an Ethernet switch to any LAN <--> port on the Time Capsule and have as many ports as you need. Ethernet switches come in 5-port, 8-port, 16-port etc versions. The switch does not need to be located near the Time Capsule, it can be located anywhere you want. That might make wiring easier. Example of an Ethernet switch:

Assuming I am able to connect the Extremes to the Time Capsule via ethernet, should their settings be "Create new network" (using then same network name and password as the Time Capsule's, of course), or should it remain "Extend Existing Network" as they are now?

"Create a wireless network" is the correct setting. "Extend" is used only if the connection between AirPorts will be wireless, not Ethernet.

Is the ethernet connection between the X1 and the Time Capsule likely to be the cause of this problem?

No, If the Ethernet connection were not working correctly, nothing would be working on the Time Capsule / AirPort side of the network.

Or is it more likely the issue is that they are so close and creating different wifi networks?

No, as long as they are creating different networks, you can have them as close together or as far apart as you want.

Would changing the X1's network name to the same as the Time Capsule's (or vice versa) fix the problem?

No, it would create more problems, since you never want to have two routers in close proximity to each other broadcasting the same network, because this will create interference effects between the two devices. Only consider this if you can locate the Time Capsule at least a room or two away from the Xi router.

I have three other AirPort Extremes (latest version) that are located in my home office, the garage, and the pool cottage. These three Extremes all extended the wifi network created by the Time Capsule. All my Apple computers, iPads, cell phones, Apple Watches, etc., connect to that network, not the one created by the X1.

Hopefully, the three Extremes are all connecting back to the Time Capsule using wired Ethernet cable connections. Is that the case?

If not, and these devices are trying to extend the network using only a wireless connection, you might not be aware that you will lose at least half of the potential speed on the network when things are set up this way. This might be OK with only one AirPort extending the network but with 3, there is a tremendous loss of signal speed and network performance. That would explain the difficulties that you are describing.

Further, in order for the three Extremes to be extending the network, they must all connect directly to the Time Capsule.....not to another Extreme.

Open up AirPort Utility on your Mac and take a look at the network layout. The Time Capsule will be located just below the Internet "globe" and all 3 AirPort Extremes should be lined up side by side horizontally so each one is connecting directly to the Time Capsule via a dotted line representing a wireless connection.

What does the layout look like if you are trying to extend using only wireless connections?

The good news is that each of the AirPort Extremes is extending the network. The bad news is that the potential speed and performance of the network has been cut to a fraction because of the 3 "extending" AirPorts.

I doubt that it will help, but if possible try moving each of AirPort Extremes a bit closer to the Time Capsule and further away from each of the other AirPort Extremes.

In addition, I would try using the Google DNS Servers on at least the Apple portion of the network and on the Xinfity side as well.

8.8.8.8

8.8.4.4

For IPv6.......set up the Time Capsule and each of the AirPort Extremes using a setting of Link Local Only.

Apple recommends a "roaming" type of network for extension, which means that "remote" AirPorts connect back to the "main" AirPort using a permanent wired Ethernet cable connection.

The best investment that I ever made in terms of network performance......far better than any new router or new computer.......was to run Ethernet cabling to major portions of the house and connect extenders that way. The difference was truly dramatic.

Happy to help, good luck!