No problem. Single lamp 120 (127V) starters run just fine even in 220V single lamp circuit with 18/20W tube.

Is it also safe to operate a 100v 50Hz high power factor HX autotransformer ballast on 120v 60Hz mains supplies as well?

If it is inductive only (no capacitors), it will mostly work. Problem could be in theory with higher OCV with lamps relying on drooping ballast output current when the arc voltage is rising, like thermal stability of saturated vapor lamps. But 20% difference does not seem to me as that big from this perspective.

Even in this case, would a power factor correction capacitor (no series capacitor) cause ballast overheating and failure if a 100v 50Hz ballast is operated on a 120v 60Hz supply? The high power factor lag type 100v 50Hz ballasts that I am considering importing have an integrated power factor correction capacitor.

I am considering the following ballast because it seems like it might be my closest solution to running North American 1000w H34 mercury vapor lamps on a 120v 60Hz supply without using any transformers at all and Japanese HF1000X mercury vapor lamps are electrically identical to North American 1000w H34 mercury vapor lamps:

https://www.jauce.com/yahoo-japan-shopping/item/lamps_22031/

This one is a 100v 50Hz high power factor HX autotransformer ballast. I might be seeming to go for this one because the low power factor 100v 50Hz HX autotransformer ballasts have been discontinued and are harder to find and I mostly stumble across 200v ballasts that are either 50Hz or 60Hz when looking at Japanese HF1000X mercury vapor ballasts and I don’t want to have to pay extra costs to get a step up transformer just to get a ballast that can properly run North American 1000w H34 mercury vapor lamps.