What do Young-Earth Creationists think about the Proto-Afro-Asiatic language? Did it really exist? And, if so, when?

All historical linguists agree that Proto-Afro-Asiatic was spoken way earlier than Proto-Indo-European.

First off, this isn't true, or is at best a) contested, and b) based heavily on dating assertions which YEC doesn't agree with in the first place. Thus, the very premise of the question is already disingenuous. (In fact, a lot of the PAA claims look suspiciously like wishful thinking trying to add credence to the incorrect assertion that humans are hundreds of thousands of years old and originated in Africa.)

The earliest attested Afro-Asiatic languages (Egyptian and Akkadian) were attested very early and they were not closely related. Proto-Afro-Asiatic had to be spoken thousands of years before [the Narmer Palette and the earliest Akkadian inscriptions].

This is completely and utterly false, and it's not even clear which of several possible incorrect assumptions are being made that permit the above conclusion. At Babel, every extant language family came into existence from a single language literally overnight. The idea that language families need thousands of years to diverge denies Babel a priori and is not supported by the inability to reconstruct a single root language. The evidence supports Babel, and Babel clearly attests to a rapid origin of widely divergent languages. The only remaining means of substantiating this claim would be to insist that it took people thousands of years to rediscover writing... and that's just silly. A determined individual, already familiar with the idea of writing (as was the case at Babel), could easily devise a new writing system in a matter of months.

Your link shows you're already at least somewhat familiar with the issues of "mainstream" dating, but on a related note, it's worth pointing out that "mainstream" consistently interprets the available data in a way that maximizes age, since this is consistent with, and nearly required for, making said data fit their overarching chronologies.

Moreover, with respect to languages specifically, there are actually a number of problems with any dating extending back closer to ten thousand years, some of which you can read about here.

Do the Young-Earth-Creationists believe that Proto-Afro-Asiatic language existed?

Frankly, this is unclear. The issue is that such languages are suppositions rather than established entities, and, while it's almost certainly true that some languages are the result of post-Babel divergence, the people that invent these "proto" languages are usually dedicated to linguistic Common Descent. Since their operating premise is false, it's difficult to be certain how far back reconstruction of common root languages can be taken until one hits the 'hard wall' of Babel. The situation is further complicated because God, at Babel, might have Created several languages with enough similarity to appear related.

For further reading, see Making some sense of Babel and afterwards.

To the extent it has been identified as a root of existing languages and/or we have written samples of the language, it is likely that something which might be named "Proto-Afro-Asiatic" existed, yes. Whether it represents a precursor to other language families, or how it fits with other languages chronologically, is very much another question.

If so, when?

Assuming such a language did exist, only two options for 'when?' are possible. Either it represents the pre-Noahic language, or it first appeared at Babel. There are claims that some written records survived the Flood (either carried on the ark, or perhaps recorded by Noah or his family, or both), and this certainly seems plausible. I'm also aware of nothing to contradict the idea that the pre-Flood language remained one of the post-Babel languages. Alternatively, it has its origin at Babel, along with every other language family (with the possible exception of one pre-Flood language).

The pre-Flood world "perished". We have many remnants thereof (fossils), but human remains and artifacts seem notably absent and were likely specifically targeted for destruction. Since this is the case, we can have very high confidence that any human artifacts or remains post-date the Flood. By extension, since dating of Proto-Afro-Asiatic is largely based on other dating claims which must be incorrect if Scripture is true, it follows that those claims are almost certainly wrong, and one of the above-stated chronologies is correct.

Honestly, what this question seems to amount to is yet another attempt to take a poorly-substantiated claim and wave it about as "disproving" Scripture... which, as usual, falls apart on closer inspection.