You're more likely to get distortion running on low gain with something that needs high gain, it can run out of power and clip. This is more likely though on low impedance low sensitivity stuff, generally with high impedance stuff it just won't get loud.

Generally with high gain it doesn't increase harmonic distortion but rather increases the noise floor. If you hear this, it would be as a background hiss, so you might hear this for example on sensitive IEMs.

It's very unlikely you'd hear it on a 300Ω overear like the HD600.

It depends on the specific device too, some devices all the low gain is doing is putting a cap on the max voltage. General rule, high gain has higher noise but some devices it doesn't, or it's tiny, well below audibility.

I don't know about this specific dongle but it uses dual CS43131 DAC chips from Cirrus Logic. Most Cirrus Logic based dongles whether CS43131 or CS43198 (I have several) have insanely good noise and distortion numbers and you really don't need to worry about it. I don't think I've seen a bad one. More likely to get distortion from trying to run something that needs more power off the low gain IMO.