Being that Baruch She'amar fell from the sky (as said in the Baeir Heiteiv Orach Chaim 51:1), how is it that the Rambam (seder tefilos kal hashna) has a different version than what we have in our siddur: did multiple versions fall from the sky?
All prayers were composed by someone, at some point, yet for many of them we have multiple versions. Generally it's because the original version is not known and people have different traditions as to what it was; sometimes it's because the original version is not known and people make different emendations in favor of what they think it must have been. I don't know which of those is true of "Baruch sheamar" (perhaps both are), but I don't see that its having fallen from the sky makes it any less subject to later changes than any other prayer.
Baruch SheAmar was not instituted by the Anshei Knesset HaGedolah, but rather by the Geonim. And it was composed by human beings, rather than falling from the sky.
The source for it falling from the sky and being instituted by the Anshei Knesset HaGedolah is the Or Zaruah, Rabbi Yitzchak ben Moshe of Vienna, in 1260.
Saadia Gaon instituted that Baruch sheAmar be recited on Shabbos, but in France, the custom developed to recite this prayer daily.
The Pri Chadash wonders at how the Geonim could institute a new tefillah. Thus,
Yet others point to early sources for it. Thus,
I've heard that sometimes 'Yerushalmi' is used to refer to midrash. But regardless, just because the Or Zarua says it fell from heaven and was established by the Anshei Knesset Hagedola does not make it absolutely so.
And at the least, Rambam (who predates Or Zarua) did necessarily hear this (later) Or Zarua. If this was not the universally accepted theory, then people would feel free to modify it.
In terms of בפי vs. בפה, see my post in which I show that there is a regular change because "fi" had negative connotations in French. And then kabbalists likely came and bolstered this change with kabbalistic ideas. But the original is the grammatical one, בפי.
In brief, this link discusses why and how some versions use the word בפי ('in the mouth of...') and some (most, actually) use the word בפה ('in the mouth,' with no grammatical connection to whose mouth it is) even though this represents a later change and is ungrammatical. It seems to be based on a kabbalistic interpretation having to do with gematria. The switch to בפה makes the word add up to same gematria as the number of words in Baruch Sh'amar, which is 87 and equal to פז, paz, refined gold, which, in turn, is symbolic of Hashem's glory.
This discussion also addresses the topic of why the text was allowed to be changed even though it "fell from the sky." Of course, many poskim feel that the text was actually instituted by the Anshei Kenneset ha-Gedolah. The discussion in the link recounts the entire history of these opinions.
As for the differences between the order of the initial praises regarding Divine Speech and Creation, I couldn't find a source for this but I would imagine it has to do with interpretations of the process of Ma'aseh B'reishit. Nusach Ashkenaz begins the process with praising the act of Creation itself, while Nusach S'fard begins with praising the the fact that Hashem speaks and creates. It's a difference in emphasis of the elements of the process of creation.