The upstairs modem is not spare. It is the main cable modem. The downstairs ASUS router is connected to that via in-wall ethernet
You can only connect one router to a modem, so it is either the Asus or the TC.
I bought the TC 3-4 years back and the AE a little before then. They should have a life more than that, no?
5 years, on average, is considered to be about the useful life for a router designed for home use. Some go longer, some head downhill sooner. I got 8 years from one Airport router and about 2 with another, so things can vary.
I have been connecting/etherneting TC to downstairs ASUS router, but if needed, I can try and connect to upstairs modem - if that helps me to utilize it better somehow
You likely have the Asus where it is for a reason, so I don't know what the advantage would be to have it close to the modem.
Leaving the TC connected to ASUS permanently and not using the AE is the last resort, but I can go that route if nothing else works.
If you want to use the TC, it must be connected permanently to the Asus router using an Ethernet connection. No exceptions to this. The only possible variation.....if you cannot run the Ethernet cable......would be to try to use Ethernet over powerline adapters to send the signal over the AC wiring in your home. I don't recommend these devices, but they seem to work for some users who are willing to accept a huge speed loss.
If I have to, how can I turn off the WiFi on the TC?
Change the setting under the Wireless tab in AirPort Utility from "Create a wireless network" to "Off". This is what the setting looks like on my TC:

If I have to, how can I 'ignore the TC wireless NW' (or use it as a guest NW) - my Mac kept picking it up last time, even though I removed it from the list of available networks. Mac connects to the ASUS network for some time, and the suddenly it switches to the TC network and then stops working. :(
You ignore it by not connecting to it. If you remove the TC network from the list of networks that the Mac has joined in the past, you have to restart your Mac for the change to take effect. Otherwise, the network will keep appearing on the list of networks. If you want the Mac to connect to the Asus network as a priority, the Asus network has to appear at the top of the list of networks. If it's not at the top, drag it there and remember to restart the Mac after you make the changes.
Unless you really do need a guest network.......(the Asus router might already offer that option).......my advice would be to turn off the WiFi on the TC and use it only as a backup drive. Then you don't have to worry about which network a device might connect to. Bottom line....... the simplest solutions always work best.