Warning: I may be wrong about some of the details here. Edits and fixes would be welcome.
Most basic features of Kinky can be used by just installing the base game and Kinky Mod: Sex, whoring, becoming a callgirl/escort, and many more. Some features require an expansion.
Ambitions - Nothing AFAIK
Pets - Required for Kinky Zoo Lover TV channel; kinky interactions with dogs and horses.
University - Required to run/register a high school; sex with plantsims.
Generations - Needed for your sim to be able to hire an NPC callgirl through the phone. However, the Generations expansion is not required for sims in your household to prostitute themselves and become a callgirl.
Island Paradise - Nothing AFAIK
Late Night - The advanced bars (the ones that can have a Barkeeper) can be used to slip drugs into drinks (starting with Patreon build 327 IIRC); dance floor (high school/prom room).
Seasons - Dance floor (high school/prom room).
Showtime - Nothing AFAIK
Supernatural - Required for the Horny Curse spell
World Adventures - Required to use the dildo training dummy; woohoo skill challenge to sleep with foreign sims.
Into The Future - Required for sex with plumbots.
Sex with Animals - Pets
Sex with Plumbots - Into The Future
Sex with Plantsims - University
Hire a Callgirl - Generations (dancer service)
Adding Drugs to a Drink - Requires a bar from Late Night
Sex, Prostitution, Exhibition, Underwear, Rapists - Base game compatible
High School - University
For a more detailed description, see http://onikikay-kinkyworld.wikia.com/wiki/Highschool
You need University for the whiteboard and podium, a buffet table (advanced version can be found in Island Paradise), and you will need either Late Night or Seasons for a dance floor. Here's what you need for each room to work:
School Classroom: Dining tables with chairs for students, a whiteboard (University), a podium (University), and a desk with a chair for the teacher.
Restroom(s): Toilets and sinks.
Lunchroom: Dining tables with chairs and a food source such as the [food] shopping register (World Adventures), advanced bar (Late Night), or buffet table (advanced version found in Island Paradise) and a dance floor.
Prom Room: Music source, dance floor (Late Night/Seasons), tables, chairs, buffet table (advanced version found in Island Paradise).
Principal's office: Desk with chair.