Alright, just going to start brainstorming the random name/restriction roulette--feel free to jump in with any thoughts.
As I figure it, restrictions should have uniform word types, and a 'challenge run' should have a set format--say, for example, we made genre of origin be adjectives (or some other category that isn't enough of a restriction on its own), sex acts be verbs, and body types be nouns. (All on a chart with numbers from 0 to 9, rolled on either on an actual die or a random number generator.) Depending on difficulty (or your cruel whim), you could use the same chart to make all your brothels "The Adjective Noun", "Adjective Nouns of Verbing", "The Verbing Noun", or even "Verb my Adjective Noun like a Noun" if you had multiple 'noun' restrictions (after all, we'll likely have multiple charts).
Or, in other words, your sex act is "oral" (verb "kiss"), your genre is "fighting game" (adjective "aggressive"), your body type is "stacked" (noun "mountains"), and the fourth roll for those who feel froggy is "red hair" (noun "ginger"). Depending on your difficulty, those rolls could be "The Aggressive Mountain", "Aggressive Mountains of Kissing", "The Kissing Mountains", or "Kiss my Aggressive Mountains like a Ginger", using the above examples.
Basically, your brothels are named and regulated by randomized mad lib.
The basic reason why I'm looking at a 'mad lib' structure is because it allows us to assign certain categories as dependent (adjectives need a chosen noun, adverbs need a chosen verb, and verbs are just bloody weird in a name without a noun), and makes it easier to shuffle in multiple types.
So, my thoughts: Adjectives (less restrictive, needs a noun) could include genres (fighting games, action series), media (video games, anime), and some broad origins (purchase, recruit, catacomb).
Verbs would be tied to actions, and would likely refer to either specific acts, clusters of acts (foreplay only, 2p intercourse only, mean stuff only), or positions allowed to be filled (bar only, hall only, club only, etc).
Nouns would be specific character restrictions. Body type, hair color, skin tone, set-specific attributes (celeb packs, porn star packs, pokemon packs), etc. These would likely need more than 10 options, for those who don't have the character packs to support certain restrictions, so that they could shuffle to their own tastes and/or sub out options they don't have.
Not sure where challenges would go, though (IE no matron, no cleaner, all girls live in luxury, no anti-preg, half girls on customer service, etc). Thematically, verbs are the restrictions, but these aren't associated with actions. It doesn't feel like it belongs with adverbs' 'thematic restrictions', either. I'm thinking maybe descriptive-nouns, like "bedlam", "filth", "luxury", "fertility", "helpfulness", etc?