Wondering if the Time Capsule has to *also* be plugged into my modem/router for this to work?
No, but if the Time Capsule is normally connected to your modem/router, things will be tons easier, and there will be much less chance of errors, if you leave the Time Capsule connected to the modem/router.
Or will the computer find/recognize the TC if it's plugged into the Mac via ethernet cable and backup just fine?
No, if the Time Capsule has been previously connected to the modem/router. If you disconnect the Time Capsule from the modem/router, then you'll have to totally reset the backup path from your Mac to the Time Capsule using Time Machine Preferences.
And...your Mac will not be available for anything else during the entire time that it is backing up to the Time Capsule.
will it continue to work as a backup disk if I later plug TC back into router and attempt to use as a traditional wifi backup?
No, not unless you go back into Time Machine Preferences and reset the backup path from your Mac to the Time Capsule.