In Japan, it’s considered slightly gauche to do this. It’s the kind of thing expected of a “country” person, or an oji-san (old man). I don’t think anyone will find it offensive or disgusting. The vast majority of people won’t care at all. And, if you are a westerner, you’re not expected to follow (or even be aware of) Japanese customs, anyway.
In China, it’s perfectly fine.
I don’t now about Korea, Thailand, India, Khazakstan, or other Asian countries.
Yes, it is correct and fine to wipe your hand, face, neck with it, be it with hot or cold towels. You can see that pretty much everywhere in Asia.
In Japan, you will be given one at hairdressers to specifically to wipe your face with it.
Specially when it’s hot and humid outside, a cold towel on your neck is something very nice!
Credit:stackoverflow.com‘