how the coin battery operates with battery backup

1. No.
2. As long as the UPS is supplying power from whatever source, as far as the Mac is concerned, it is no different than being plugged into any other power receptacle. Once the UPS battery is depleted, it's as though the Mac's power cord has been disconnected.

I don't think the NVRAM batteries (there are two) will be affected either way. They ought to last on the order of ten years or so — the shelf life of the battery.

If I have the iMac turned off but plugged into a fully-charged, external battery (UPS) that is also unplugged, will the iMac use the coin battery....

Yes it will, and use it a lot. I killed off the internal backup battery in about year doing exactly what you describe with an older Mac. Learned my lesson.

When the computer is on wall power but turned off, a small trickle current bypasses the power supply to maintain parameters. The coin battery is to back up that bypass system should you need to unplug the computer for service or a move. If you turn everything off with the battery backup or a power strip, the coin battery kicks in.

Given that, since 2012, Apple has not sold any iMacs designed to be opened for battery replacement without a service provider, it behooves you to respect the system. Our iMacs are on battery backups with surge protection, and the backups are turned ON and plugged in all the time. That way the internal backup battery can remain doing its intended function: short-term powering to maintain parameters. Yes, we get nasty t-storms, but a quality backup power supply should be able to handle that.

The good news is that since the "New World ROM" Macs appeared in the late 1990s, a dead backup battery does not cause anywhere near the grief it did in "Old World ROM" Macs (basically those before built-in USB ports).

Not for the purposes of maintaining the NVRAM batteries; no.

There are other valid reasons for using a quality UPS though. Thunderstorms or power anomalies can't always be anticipated.

The batteries are not rechargeable. They will degrade at their normal rate regardless.

It's actually simpler than that. Many UPSs use USB communication to report its status to the Mac, and tell it when utility power is lost. That way, macOS can provide an informative dialog. If you are not present to see it the Mac can perform an orderly shutdown on its own. All those features are incorporated in macOS. Read Set when your Mac shuts down while using a UPS - Apple Support.

If not for that feature desktop Macs would have no way of knowing a power failure has occurred.

Having said that, if you are present when an electrical storm is likely, I agree the best thing to do is to disconnect anything and everything of value from utility power. All quality UPSs have built-in surge suppressors, and all public utilities incorporate lightning arrestors at various points in their power distribution networks, but literally nothing can provide absolute protection from the virtually limitless energy a nearby or direct lightning strike can inflict upon residential or office wiring.

Hi,

The coin cell battery, you refer to, otherwise known as a CMOS 2032 battery, typically can cause the system to lose its date and time reference when there is no power to the system, though if you set NTP(probably it's a Apple(time.apple.com)), it's automatically set proper time when start up your iMac.

Yes ...

Obviously the UPS battery is rechargeable.