etymology - "cathouse," "call house," and "sporting house" for "bordello"

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

cat:

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.

Answer

The term cat has been used to express contempt for a human being in the past centuries. OED cites early usage examples:

• fig. a. As a term of contempt for a human being; esp. one who scratches like a cat; a spiteful or backbiting woman. spec. an itinerant worker (U.S. slang)

  
  
  
  
  
  • 1225 Ancr. R. 102 Hweðer þe cat of helle claurede euer toward hire.
  
  

from which probably the usage to refer to a prostitute.

Cathouse:

• "... according to the OED the
  now-obsolete use of the word "cat" for "prostitute" 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." ".

According the New Patridge Dictionaty of Slang the term cathouse meaning brothel in US is from 1893:

• Cathouse noun a brothel US. 1893 • She looked as if she might have worked half those years in a cat house.

Euphemisms are common to refer to a brothel mainly for cultural or legal reason. Call ( as in callhouse) may suggest the idea of paying a "visit" to someone, a nice euphemism.

• A brothel is a place where people may come to engage in sexual activity with a prostitute, sometimes referred to as a sex worker. Technically, any premises where prostitution commonly takes place qualifies as a brothel. However, for legal or cultural reasons, establishments sometimes describe themselves as massage parlors, bars, strip clubs, houses of ill repute, body rub parlours, studios or by some other description.

According to The Dictionary of American Slang, (Fourth Edition by Barbara Ann Kipfer, PhD. and Robert L. Chapman, Ph.D.):

Call house

• A brothel, esp one where a prostitute may be engaged by telephone : that call joint (1910+)

The same origin is suggested also by the The Routledge Dictionary of Moder American Slang:

• Call house (noun): a brothel from which prostitutes are procured by telephone US, 1913 •