Glenn Kempf wrote:P.S. ...On the other hand, nearly all of the PE words in the lexicon beginning with "f" begin with "sh" in Axunashin, in accordance with the regular sound change. Perhaps this is the real correspondence?
Yes indeed; the f for Axunashin is just wrong. (Or perhaps just outdated. I wrote that part very early, and the data may have changed later.) I'll correct that section.
Glenn Kempf wrote:(And how would someone reconstructing PE know which of the modern sounds (here, f as opposed to b, b/v, sh, or h) is/was the "original" one? Probably simply through exhaustive comparison...am I right?)
We can't know for certain; we're really just making our best guess. Some historical linguists consider proto-phonemes to be just a shorthand for the correspondences-- e.g. if we reconstruct *f here, the important claim is that there's one phoneme here, not that we know for sure that it's a bilabial (or labiodental...?) fricative.
Of course, we also try for the most phonetically plausible reconstruction that's compatible with the evidence. Some changes are a lot more likely than others; other changes, though equally possible, don't accord with the facts as well.
In this case, *f is probably the best choice, though not necessily the only one you could defend. *h is very unlikely: f

h is a well-attested lenition, while h

f is not; also *h doesn't relate to the labials in the other languages. The latter point also rules out *sh; f

sh and sh

f would seem to be equally likely, but *sh couldn't easily explain the labials in Cu?zi and Cadhinor.
*v is not very likely-- f

v is a common lenition, and we'd have to have some extra evidence to posit a v

f fortition instead.
That brings us to *b. The posited f

b is a fortition. It's not that uncommon to have fortitions in initial position, and positing it makes for a lot fewer assumptions in general (none of the other languages have a b, and even Cu?zi doesn't have it in all environments.)
So, that leaves us with *f. Of course, one could also argue for another sound entirely that led to all these correspondences... perhaps *p. In this case I don't think *p offers any explanatory advantages over *f, and since it's not itself attested, I'd rather go with *f, which is the most widely attested reflex anyway.
Historical linguists argue long and hard over the precise details of proto-languages. There's about four competing reconstructions of Old Chinese, and even in Indo-European, there are deep disagreements over the stops and the laryngeals.
Another level of disagreement comes in when people attempt to compare protolanguages. E.g., people reconstruct Old Chinese rather differently if they base it only on the Sinitic dialects (e.g. "Old Chinese" on my numbers page), or on information from proto-Tibetan as well (e.g. "proto-Chinese").