All three terms appear to be euphemisms for house of prostitution and are marked as Americanisms by Robert-Collins French and English Dictionary, Ed. 1985.
cathouse being the most common one (as shown in this Ngram), I was wondering, what actually is implied by "cat"?
Is it, as stated here, because such establishments would place a statue of a sitting cat in one of their windows, facing outward when open for business, and facing inward when they were closed?
Or is it on account of the now obsolete use of "cat" for "prostitute" which, according to OED, was established as early as 1401 when the word appeared in Friar Daws Reply, a poem in Middle English?
Although the exact meaning of the word in this poem is debatable, the definition had been unquestionably cemented by 1670, when an early dictionary entry for "cat" followed the word with the phrase, "a common whore." Narkive Newsgroup Archive
As a term of contempt for a woman, from early 13c. Slang sense of "prostitute" is from at least c. 1400. Slang sense of "fellow, guy," is from 1920, originally in African-American vernacular; narrower sense of "jazz enthusiast" is recorded from 1931. Etymonline
Or, is it in relation to "cat" as a possible older slang word for female genitalia, sort of analogous to modern day "pussy"?
In addition, do the terms sporting house and call house, aside from being rare euphemisms for "brothel," have a somewhat antiquated feel to them, sort of like the word "saloon" might conjure up the picture of an Old West barroom?
sporting house: bordello; a long career as a madam in a New Orleans sporting house.
First Known Use of sporting house: 1615 M-W
According to Wentworth & Flexner, call house originally denoted any brothel and call girl any prostitute working in a call house. (Presumably call girl was in some sense an abbreviation of call house girl.) Sense Developments
That "call house girl" sense of call girl is also supported by the Oxford Dictionary of Euphemisms, which quotes:
A call girl or call-button girl was not originally someone whose attendance was requested over the telephone but a prostitute who lived in a call house, where men might visit or call.