Early sources cited in the RHHDAS
J.E. Lighter, Random House Historical Dictionary of American Slang (1994) has a succinct entry for geetus:
geetus, n. {orig. unkn.} money. Also var[iant]s. [Earliest three citations:] 1926 Finerty Crimanalese 25: Geetus—Bankroll. 1935 Pollock Und. Speaks: Gheetus, money. 1936 Duncan Over the Wall 113: I wanted some cash, geetas, dough-ra-me, a bunch of the green fodder that makes the world go round.
Finerty is James Finerty, Criminalese: Slang Talk of the Criminal (1926), which is essentially a 72-page pamphlet, published in Washington, D.C. by the author. Julie Coleman, A History of Cant and Slang Dictionaries: Volume III: 1859-1936 (2009) says of the author:
Finerty is described on the front cover as an 'Ex-Chief of Police', but a list of his best-known cases does not reflect well on him. It includes the Mary Phagan murder trial, a notorious anti-Semitic miscarriage of justice in Atlanta, Georgia, and the arrest of the Ashley-Mobley gang in Florida.
To judge from this smattering of details, Finerty had no strong connection to California. He took many of the entries for his glossary from Joseph Sullivan Criminal Slang (1908)—I don't know whether geetus was among them—but Sullivan was (again according to Coleman) a bail commissioner in Suffolk County Massachusetts and a member of the Boston Bar.
Pollock is Albin Jay Pollock, a veteran of the Spanish American War. The Underworld Speaks (1935) The Find a Grave website reports that Pollock was born in New York City in 1876 and died in Yountville, California. An included obituary excerpt from the Arizona Republic offers this information:
Pollock, a practicing geologist for 40 years, lived in San Francisco most of his life before coming to the veterans home [in Yountville] in November, 1966.
Perhaps the most interesting thing about Pollock's use of gheetus is the spelling—the point of starting it with the letters gh seems to be to indicate that it is pronounced as a hard g rather than with a j.
Duncan is Lee Duncan, Over the Wall (1936). He is identified on the cover of his book as "Ex-Convict No. 9256, Oregon State Prison," and Eric Partridge, A Dictionary of the Underworld: British and American, third edition (1968) says that "Duncan himself spent many years" in the Oregon State Prison. A fuller version of the quotation from Duncan's memoir runs as follows [combined snippets]:
Frisco's Embarcadero, after so long an absence, looked both familiar and good to me and I did not have the guts to leave—to place more terra firma between me and the Oregon stir. I went back to the stick-up racket, for I wanted some cash, geetas, dough-ra-me, a bunch of the green fodder that makes the world go round.
So there Duncan is—in San Francisco, pulling armed robberies for geetus.
***Coverage of 'geetus' in 'Dictionary of American Slang' and elsewhere
Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition
(1960) is the predecessor of Barbara Kipfer & Robert Chapman, Dictionary of American Slang, fourth edition (2007), whose definition is rather misleadingly rendered by Dictionary.com, as cited in the posted question. Wentworth & Flexner offers this entry for the term:
geedus geetis geetus n. Money. c1935. Underworld use. 1943: "The pitchmen must give the store a 40% cut on the 'geedus.' ... Zolotow, SEP, 13. Pitchman use.
Zolotow is Maurice Zolotow, Never Whistle in a Dressing Room; Or, Breakfast in Bedlam (1944), although the quoted language first appears in an article in the Saturday Evening Post, volume 216 (1943). Here is the SEP's version:
The tip, in pitchman argot, is the crowd, and turning the tip is the most important phase of the scenario, when the pitchman has finished revealing the wonders of his little article and is about to crack the price and extract the quarters or dollar bills from his tip.
Gasoline restrictions, on the one hand, have driven many pitchmen to holing up in five-and-dime stores or department store bargain basements for the duration. The pitchmen do not like this, as they must give the store a 40 per cent cut on the "geedus," or money. As a rule, they prefer to work on a 70 or 80 per cent mark-up. The five kitchen gadgets, for example, which Louie sells for a dollar cost him from twenty-five to thirty-three cents.
Zolotow's book then reintroduces the term geedus eight pages later:
As soon as you have them laughing, you know it will be easier to get that geedus out of their pokes. Usually, if you know your business, by the time you are ready to rack the price there will be someone who is reaching into his poke, and then you work directly at him, and as you finish you reach for the merchandise and duke him, that is you hand him the article and you say thanks.
Clearly geedus here means simply money—there is no sense that the pitchman is "reversing or altering traditional money." And the entry in Kipfer & Chapman, Dictionary of American Slang, fourth edition (2007) says nothing about money reversal or alteration either:
geetus or geetis or geets or geedus n. Money ; Pitchman must give the store a 40 percent cut on the "geedus"/ "I'm spendin' my hard-earned geets (1930s+ Underworld & hawkers)
Hyman Goldin, Dictionary of American Underworld Lingo (1950) has this brief but suggestive entry:
Geedus. (Pacific coast area) Money.
Eric Partridge, A Dictionary of the Underworld: British and American, third edition (1968) has this entry for the term:
geetas or geetis. 'The criminal gas a slang term for almost every denomination of money. Money in general, he refers to as dough, cash, mazuma, jack, geetis, and sometimes diners,' Convict, 1934: ... extant. Origin? Perhaps a corruption of gelt.
None of the other dictionaries I consulted second Partridge's etymological suggestion. However, An interview with writer S.J. Perelman reprinted in Writers at Work: The Paris Review Interviews, volume 2 (1964) [combined snippets] contains this intriguing juxtaposition:
Interviewer: Feeling as you assert Mrs. Parker and Mr. Benchley did, and as you plainly did, how could you manage to remain there [in Hollywood] for even limited periods?
Perelman: We used, to ask each other that with great frequency. The answer, of course, was geetus—gelt—scratch. We all badly needed the universal lubricant, we all had dependents and insurance policies and medical bills, and the characters who ran the celluloid factories were willing to lay it on the line. After all, it was no worse than playing the piano in a whorehouse.
The Yiddish dictionaries I consulted include entries for gelt but none of them even mentions geetus, which leads me to think that Partridge is simply speculating about its possible etymological roots.
As noted in the posted question, Jonathon Green, Slang Dictionary (2008) suggests a different etymology, but without much conviction:
geetus n. (also geedus, geetas, geeters, geetis, geets, ghedis, gietus) {ety. unknown; ? SE get us {1920s+} (US) 1 money. 2 power.
Clarence Major, Juba to Jive: A Dictionary of African-American Slang (1994) has this entry:
Geets; geetus n. (1940s) power or money or both. ... S[outhern and] N[orthern] U[se]
The late date of claimed origin in usage suggests, however, that the term migrated into African-American slang, rather than originating there.
Other early matches for 'geetus'
Google Books and Elephind searches for geetus and its variants turn up several matches from the 1920s and early to middle 1930s.
The earliest match that a Google Books search for geetus turns up is from Herman Schnitzel, Unt Comes It Now (1928) [combined snippets]:
...langwitch of per cents unt discounts. It aint a very musical language unt iss best schpoken to der accompaniment of much gestures mit der hands.
Der Hebrew language died of overwork, but its motto still lives in der hearts of its people, "Get Der Geetus."
This ludicrous exercise in ethnic humor is not to be believed on any level, but it does have a somewhat surprising close link to California. An Amazon listing for an LP recording of The Best of Professor Schnitzel (1964) offers this background on the ersatz professor:
Clarence Coleman, a San Francisco realtor, developed the character of Professor Herman Schnitzel for the Blue Monday Jamboree on KFRC in 1927, and later performed as the same character on KYA and the ABC Western Network in 1929. The Professor was known as "Der Feetsball Foolisher", "Doctor of Nonsense" at the Oooniversity of Cincinnapolis. His catchphrase "Unt Comes it Now" was repeated by radio listeners up and down the coast. Professor Schnitzel did a morning show on WHUM Radio, Reading, P[ennsylvani]a in the 1950s & 60s and did live comedy in and around Reading, Pa.
From Gus Vignolle, "Sports Vignettes," in the [Santa Monica, California] SaMoJaC (September 28, 1932):
Seen and heard while meandering here and there: ... Another ex-student, Sam (the Kibitzer) Katz, is making geetus while on his tour with the carny (carnival to you)...
From a glossary of underworld terms in The Editor: The Journal of Information for Literary Workers (1932[?]):
Geetus: Money; bankroll. "You gotta have the geetus in this racket."
The Editor was published in Highland Falls, New York.
From a piece of fiction in Collier's, volume 91 (1933):
"No actor never gave nothing to nobody," I said, punching it up with a good old prop razz.
"This actor did," replied Skunk. "You never worked with Georgie. They don't make 'em any better. But good or bad, Buff earned the geetus, and I'll tell you why."
From "The Letters of Bill Hailey," in the Sausalito [California] News (June 25, 1937), a letter directed to "Hon. A.H. Samish, Kingpin Lobbyist, Kohl Bldg., S.F.":
You should worry, Hon. Samish, as long as you get the "geetus," or whatever that favorite expression of yours used to be when you were getting your self-made education.
The whole trouble is that there I are a lot of birds jealous of you just because they figure they can ; horn (I don't mean Benny) into the picture within a few years or so and expect to learn all the answers or draw rabbits out of a hat or whatever it is that you are supposed to do.
Conclusions
The foregoing research suggests provisional answers to the first two posted questions and a fairly definitive answer to the third.
Speculation that geetus derives from get us (Green) or gelt (Partridge) seems to have come from the thin air. None of the sources I checked offers any support for either hypothesis.
Early published instances skew heavily toward California as the term's geographical place of origin. Including the Oakland Tribune article cited in the posted question, we have nine instances of geetus from the period 1926–1937; of those, six (from 1926, 1928, 1932, 1935, 1936, and 1937) have definite connections to California (and many of them more specifically to San Francisco), one (from 1926) is most closely tied to Washington, D.C., one (from 1932) comes from upstate New York, and one (from 1933) appears in a national magazine without more specific geographical clues. The circumstantial case for California origin is strong but (owing to that very early 1926 instance from Washington, D.C.) not overwhelming.
The Dictionary.com citation of Kipfer & Chapman, Dictionary of American Slang, fourth edition (2007) is erroneous. I own the book in question, I have checked the entry for geetus, and the definition "A person who tends to reverse or alter traditional Money" appears nowhere in it. I have no idea where Dictionary.com obtained its definition. Google Books returns no matches for the wording it gives, and a general Google Internet search returns only the Dictionary.com entry for geetus and three copies of that entry posted on another dictionary site. In any case, attributing the quoted language to Kipfer & Chapman is incorrect.