From 'The Insect That Stole Butter' - the excellent Oxford Dictionary of Word Origins.
Under Raj:
"The 1930s slang word of approval 'tickety-boo' has no association with tick or ticket, but probably comes from a Hindi expression thik hai - all right."
It seems The Walrus got the answer 8 years ago.

Come a Cropper.
To come a cropper is to suffer a defeat or disaster. The origin of the phrase maybe the 19th-century hunting slang term 'cropper', meaning 'a heavy fall'. Cropper probably came from neck and crop, an expression meaning 'completely or thoroughly' and originally used in the context of a horse falling to the ground. Crop here referred either to the rider's whip (originally the top part of a whip) or the horse's hindquarters. This sense is found in Old French croupe 'rump', which appears as croup in Middle English, and is the source of the crupper [ME], the bit of harness that goes from the saddle under the horse's tail, and which lies behind the word croupier [E18th]. In early use, this was a term for a person standing behind a gambler to give advice, adopted from French, cropier 'pillion rider, rider on the croup'.â€쳌

Eric Partridge’s Dictionary of Historical Slang confirms this:

“Cropper; esp. come, or go a cropper. A heavy fall, fig. or lit.: from the late 1850s; coll.
Trollop, 1880, “he could not… ask what might happen if he were to come a cropper.â€쳌
Ex. Hunting.â€쳌
Jim Carroll