Can a failed harmonic balancer/damper create vibration that can be felt?
Absolutely. Some torsional dampers carry an offset weight intended to "balance" the engine. Depending on the construction of that weight--and the particular failure of the damper--the engine goes out of balance and shakes like a dog shiiting razor blades.
Or, the outer ring of the damper gets loose and wobbles around, making noise and causing out-of-balance issues because the ring is no longer centered around the hub. In which case the engine shakes like a dog shiiting razor blades.
Or the damper can have a fairly "mild" failure, the outer ring shifts but doesn't get "loose"; and nobody knows about it until the engine can't be timed with a timing light. Poor power, poor fuel economy, and the thing is only caught when someone finds true TDC and compares it to the timing marks.
My understanding is that it damps torsional vibrations across the crankshaft, removing them such that they don't appear at the flywheel end where they could be transmitted to the drive train and thus the rest of the vehicle. Is that correct?
Sort of. Removing torsional vibration is primarily of benefit to the crankshaft and bearings. Reduces cracking of castings, and premature failure of bearings.
Any number of engines have no "damper" at all--including all forged-crank 283
at least up to '64. The cast-crank 283s got real-live dampers, the forged crank units got solid hubs. Seems to me that Cadillac had solid hubs rather than "dampers" also...but I'm not totally sure about it.
Is there a balancing function also? If so, on all or only the V6s?
Depends on the engine. Far as I know, with regard to the GMT400 engines, the 7.4L/454 uses offset weight in the damper. 305 and 350s do not. And I don't remember about the V-6.