Where does 'po-faced' come from etymologically, geographically, and chronologically?

Not a complete answer, but the following extract from notoneoffbritishisms.com provides some more information about the “pot” origin assumption:

Kate Trimble Sharber (The Annals of Ann (1910)is using po-faced to mean more or less gaunt. Since this doesn’t show up anywhere else, and since the Bright Young Things most likely weren’t reading obscure American novels, I would take this as a rare, possibly unique, piece of regional dialect.

When the Lingua Franca piece appeared, the prolific and valuable language commentator Stan Carey posted a comment that the American Heritage Dictionary was a bit more definitive than the OED about etymology, stating that the term comes from pot (pronounced “po”) de chambre, French for chamber pot, “a po-faced expression being likened to that of a person observing the contents of a chamber pot with disgust.”

The OED has an 1880 citation for po-as-chamber pot (in a dictionary of the Scottish Language, interestingly), and a use of it by Leonard Woolf in a letter written in 1905 (a time when Woolf, having graduated from Cambridge, was serving in the Ceylon Secret Service): “I have to help to see that King’s House is prepared for him, to reckon out how many fishknives & pillow cases & pos he wants.”

The Oxford English Dictionary ventures a few theories in its etymology of po-faced, which is a sign that there isn't conclusive evidence for where the word comes from:

Etymology: Perhaps < either poh int. or po n.4 + faced adj.1, or perhaps shortened < poker-faced adj. at poker n.4 Compounds 2 (perhaps after either poh int. or po n.4). Perhaps compare also pie-faced adj.(Show Less)

Omitting the idea of po = poor (which isn't presented here), the options presented are:

• poh, an interjection conveying contempt, cf. pooh
• po, the chamber pot (perhaps signifying a face made when near one or carrying one
• poker-faced, meaning deadpan
• pie-faced, meaning stupid; originally American slang from late 19th century

So I don't think there's a clear way to determine the origin between these options.

To answer your other questions directly:

Is the 1910 U.S. instance of "po-faced" directly (genealogically, as it were) connected to the British slang term or are the two forms of "po-faced" unrelated and merely coincidental?

Likely coincidental. Po-faced in the 1910 example is probably an example of eye-dialect for poor-faced, or nonstandard spelling that represents pronunciation. Dropping the final consonants off words is another sign of eye-dialect. The later British quotes don't show evidence of eye-dialect, and the word means something different. So there's very little to build a connection on.

Is the 1910 U.S. instance of "po-faced" unique and idiosyncratic, or does it represent a contemporaneous idiomatic usage of "poor-faced" in the United States during the early 1900s?

Likely idiosyncratic. Eye-dialect was common in the US to represent characters speaking nonstandard dialects, particularly white and black southerners (see e.g. Mark Twain). Given the lack of Google Books and newspaper database results for the word, the American usage feels specific to eye-dialect rather than being a standalone term, at least in the early 1900s.

What is the earliest documented print instance of "po-faced" in the British English slang sense?

The 1934 usage quoted in the OED entry is certainly early:

1934 C. Lambert Music Ho! iii. 191 I do not wish, when faced with exoticism, to adopt an attitude which can best be described by the admirable expression ‘po-faced’.

I haven't been able to turn up anything earlier than that.

In Gullah — the dialect spoken by "black people living on the sea-islands and tide-water coastline of South Carolina and Georgia" (OED), including any area reached by diaspora — 'po' means "poor, also thin, lean, low in flesh" (Ambrose Elliott Gonzales, The black border; Gullah stories of the Carolina coast, 1922). So a (paywalled) 1911 newspaper review glosses "po-faced" in a passage from Kate Trimble Sharber's 1910 The annals of Ann, as "not round":

Mammy Lou disapproves of lovers whose faces are not round. "He is tolerable po-faced, which ain’t no good sign in marryin’. If thar’s anybody better experienced in that business than me and King Solomon I'd like to see the whites o’ther eyes; an’ I tell you every time, if you want to get a good-natured, wood-cuttin’, baby-tendin’ husban’ choose one that’s fat in the face!

The meaning of 'po-faced' in Gullah (and more generally in the Gullah diaspora) resembles the meaning of 'thin-faced' in broader and longer use; 'thin-faced' has been a lexical item in varieties of English since at least 1616 (Shakespeare, "Twelfth Night"). Both lexical items describe similar facial attributes. Context determines whether either one connotes anything more about those described as 'thin-faced' or 'po-faced' than the facial attributes. Mammy Lou's assessment of Mr. MacDonald's character, for example, starts uncertainly on the basis of what she describes as the leanness of his face.

The 1910 US use of 'po-faced' by Sharber may be related to a use by Constant Lambert, a British author and musician who had an "intuitive sympathy with the American negro" (Christopher Palmer, "Constant Lambert — A Postcript", Music and Letters, Volume LII, Issue 2, April 1971, 173–176). Lambert, who was familiar with US jazz and an enthusiastic admirer of Duke Ellington as well as Florence Mills (see Bill Egan's 2004 Florence Mills: Harlem jazz queen, p 91), had quite possibly encountered 'po-faced' used with the Gullah meaning modified by circumstantial connotations. In the 1934 Music Ho!, he writes that the

… trouble with exotic music is that so much of it is emotionally and technically two-dimensional. The austere exoticism of Stravinsky’s rhythms soon becomes as wearisome as the lush exoticism of Delius’ harmonies.

I do not wish, when faced with exoticism, to adopt an attitude which can best be described by the admirable expression ‘po-faced’. We cannot live perpetually in the rarefied atmosphere of the austerer classics, whether ancient or modern, and it is absurd not to enjoy a work merely because it is essentially sterile in influence. Personally speaking, if it is a question of choosing between an exotic work and a so-called abstract work, give me exoticism every time. But as we are examining in this chapter the decline of the classic tradition it is necessary to lay more emphasis on the fatality of that femme-fatale exoticism, than on her feminine charms.

The 1934 use by Lambert could be considered transitional between the Gullah (and, more broadly, southern US) meaning of "a lean face" and the later British slang meaning of "an expressionless face". However, the available evidence goes farthest to support that the contemporary British slang sense developed from 'poker-faced', which was in common British use during the early decades of the twentieth century (attested from 1915 in OED).

Lambert's 1934 use provides the earliest attestation of 'po-faced' in OED, which defines it as "characterized by or assuming an expressionless or impassive face; poker-faced; (hence) humourless, disapproving" and describes the word as "colloquial (chiefly British)". The Gullah meaning, "poor, also thin, lean, low in flesh", and the 1910 use by Sharber, are not documented in OED.

In addition to the 1934 use by Lambert, OED attests the British slang sense of 'po-faced' with two more early appearances, one from the very popular 1936 play "French without Tears" by Terence Rattigan (1982 edition), and another from a 1951 novel. I was unable to examine the context of the second, the 1951, citation.

The 1937 edition of Rattigan's popular play, the date cited by OED, prints "pie-faced" rather than "po-faced". The latter, 'po-faced', does appear in the 1982 edition linked in my previous paragraph, and is also associated with a 1938 edition at HathiTrust. In context, it is apparent that the meaning given by OED for the chiefly US slang 'pie-faced', "having a round, flat face or a blank expression", is the meaning intended in Rattigan's play:

(They both suddenly become aware that ROGERS isn't laughing. … BRIAN silently frames the word "Pie-faced" in his mouth)

ROGERS
(With a wooden face) …

Both 'pie-faced' and 'poker-faced' invoke the "blank expression". Neither conform with Gullah 'po-faced', a face "thin, lean, low in flesh". If Lambert's use in 1934 is transitional, 'po-faced' travels far to get from "a thin, lean, low in flesh face" to "a round, flat face". Given that the simplest explanation is more likely to be correct, Lambeth is more likely to have been using the British slang 'po-faced' with the meaning of either "poker-faced" or, less likely, "pie-faced", than drawing on some glancing acquaintance with, and a poor understanding of the meaning of, Gullah 'po-faced'.

'Po-faced' isn't found by OCR in The British Newspaper Archive until the last half of the twentieth century. In 1957, as a clipped compound from 'poker-faced', the Daily Herald (London) uses it twice, on (paywalled) 12 and 13 December:

12 December 1957

He told how he asked Lord Weeks, a director of Royal Exchange, if he had given anything away by his looks.

Then he added: "He said that if anything I looked more po-faced that day."

13 December 1957

A man's capacity to survive the situation would depend, he suggested, on whether he was good at poker.

And there was nothing far-fetched in this suggestion.

Lord Kindersley himself testified that a fellow director had said to him, "Thought you looked particularly po-faced that day."

Now just consider the further fact that these superhuman poker-players or po-faced superhumans, whichever way you like to put it, were not indulging in a game of tiddley-winks.

These uses in 1957 suggest 'po-faced' was established gaming slang, meaning "poker-faced". More than 1100 OCR hits on 'po-faced' in the BNA corpus for 1957-1999 follow the 1957 uses.

Early uses in the British popular press support the likelihood of derivation of the clipped compound 'po-faced' from the existing compound 'poker-faced'. Existing US slang, 'pie-faced', may have influenced the adoption of 'po-faced' from 'poker-faced', by way of semantic overlap.

Derivation of 'po-faced' from 'poh', an interjection expressing "contempuous rejection" (OED), is not supported by the available documentary evidence. Neither is derivation from 'po', meaning "chamber pot".