So, in my current setup where I have the Arris as one network and the TC (and AirPort "extenders") as another network, there's no downside to those two routers being in close proximity to one another?
That's correct.
Is there no performance issues (assuming they navigate to good channels)?
No issues, since this is the correct setup when you have two devices in close proximity both producing a Wi-Fi signal.
I think I would like that step-by-step on how to set it up as one single network where the Arris is in one room, the TC in another and they're connected by an ethernet cable
You have done a good job setting things up, so I don't think you really need step by step.
To set things up this way, you would first power off the AirPort Extreme that is now extending the Wi-Fi signal that the Time Capsule is providing.
The next step would be to simply change the name and password of the Wi-Fi network that the Time Capsule is now providing to exactly match the same wireless network name and password that the Arris modem/router is producing.
Then, you would perform a Hard Reset on the AirPort Extreme to erase the current network settings, temporarily move it close to the Time Capsule and then set up the AirPort Extreme to extend the Wi-Fi signal from the Time Capsule. When the AirPort Extreme displays a green light, move it back to the area where it will reside.
Note.....it is very important that the AirPort Extreme receives a much stronger signal from the Time Capsule than it receives from the Arris modem/router to allow the AirPort Extreme to extend correctly.
If you are willing to experiment, you could probably get away with having the Arris and Time Capsule in the same room, as long as they are not side by side or very close together. If you could use a 15-25 foot Ethernet cable to locate the Time Capsule at the other side of the room or along another wall, you will probably be OK.
The real reason that you don't want two devices both producing the same wireless network when they located close together is that this will confuse other Wi-Fi devices like your laptops, phones, etc because they won't be able to clearly detect the stronger signal between the Arris and the Time Capsule.
It would be like an FM radio station setting up two tall antennas side by side to both broadcast the same radio signal. Not only would this confuse car radios that are connected to the radio station, there would be no gain in the actual signal strength of the signal. It would make much more sense to set up the second radio antenna 20-30 miles away from the main antenna because the radio station would have much more reach that way.
Good luck!