Originally Posted by bhomburg

There's no covid-19 related restrictions on pet travel. All regular policies apply. If flights are operating, you can take pets along as usual.
One word of caution: with all these travel restrictions in place, document checks and the check-in procedure will definitely be more thorough than it used to be.
Korean has a 5kg weight limit for pet in cabin travel. With carrier and bedding, you will exceed this with a 4.5kg dog. An appropriately sized soft-sided carrier will weigh at least 1,3 kg.
Asiana has a 7kg weight limit for pet in cabin travel. I`d rather book on Asiana to be on the safe side here.

edit: Asiana looks like the better option for other reasons as well: The layover in ICN is shorter, and: Korean doesn't fly to Da Nang on their own planes but codeshares on Jin Air, which is a low-cost carrier. Even if they do accept pets in the cabin (I didn't check), you would need to book the animal separately with them, pay a second fee, and change terminals in Incheon. Really, don't do this and fly on Asiana. One airline, one stop, one pet fee, a single, quick one-hour layover with no transit woes, and widebody aircraft all the way (A350 for the long transpacific flight to Korea, then A330 to Da Nang - much better than the high-density no-legroom 737 you'll have to endure on Jin Air for almost five hours)
There's a few other threads already dealing with transiting ICN with pets.