I couldn't find any use of the phrase earlier than the 1840 Money Diggers reference, but I did find some background to which the saying might refer. Apparently the debate on cat-skinning boiled down to whether or not it was done while the cat was still alive. Here's a clip from the disturbing House of Commons' Minutes of Evidence Taken Before Committee on Bill for Prevention of Cruelty to Animals, 1832:
And here's confirmation from The Leisure Hour, 1879, that cats were used for womens' furs, but with a denial they were ever skinned alive:
So, to answer your second question, yes, it was always quite gruesome.
Summary
The American cut your stick, to die, comes from the British cut your stick, to depart, which dates back to at least 1813.
Americanisms
Uncle Tom's Cabin by American author Harriet Beecher Stowe was published in 1852 so let's check an almost contemporary dictionary.
Maximilian Schele de Vere's Americanisms; the English of the New World (1872) says on page 594:
Irish origin?
The OED has the British English sense of departing from 1825 but with no etymology.
The Wordsworth Dictionary of Phrase and Fable (2001) based on the original by Ebenezer Cobham Brewer (1810 – 1897) offers an Irish origin:
I must cut my stick--i.e. leave. The Irish usually cut a shillelah bfore they start on an expedition, Punch gives the following witty derivation:--
"Pilgrims on leaving the Holy Land used to cut a palm-stick, to prove tey had really been to the Holy Sepulchre. So brother Francis would say to brother Paul, ' Where is brother Benedict ?' 'Oh (says Paul), he has cut his stick ! ' — i.e. he is on his way home."
OED antedating
The OED can be antedated in Grose's Classical Dictionary of the Vulgar Tongue (1823, page 107 -- but not in the 1788 or 1796 editions) which simply says:
CUT ONE'S STICK. To be off. Cant.
Further antedatings
Further antedatings can be found in Othello-travestie: In Three Acts, with Burlesque Notes in the Manner of the Most Celebrated Commentators and Other Curious Appendices by John Poole (and William Shakespeare) (page 8):
Roderigo.
Why not cut your stick ? (b)
Page 29, just before Cassio leaves:
Cassio.
I'll cut my stick.
And in the extensive footnotes:
Given the subtitle of the book -- "with Burlesque Notes in the Manner of the Most Celebrated Commentators" -- I don't think we can trust these notes to be real, and therefore the 1597 to be fictional, but they are at least 1813 examples.
Best Answer
"More X than you can shake a stick at" means more than you can count. I don't know the origin but a as a wild speculation picture someone using a walking stick or cane to count something. If there's lots to count, the stick will be shaking a lot for each item. If there's too much, the shaking stick won't be able to keep up.
The OED says it's a figurative use of shake but doesn't give any more on the origin other than saying it's colloquial, originally and chiefly U.S., and giving the same 1818 as in the question.
It's originally North American, but it is now commonly used and understood in the UK as well.
I found an earlier example from 1794, but without the comparative "more X than...". British Synonymy: or, An Attempt at Regulating the Choice of Words in Familiar Conversation, Volume 2 by Hester Lynch Piozzi:
But this is British and the full phrase appears to be American, so they may be unconnected.
World Wide Words is usually a good source on these things. If they summarise: "nobody knows for sure", then that's probably the best we have.
However, there is this from alt.english.usage FAQ that questions whether the original meaning was different to today's: