The cat is going to be of some sort of importance later in the game I hope so, or is there a quest with it on this update?
No importance, and no stated plan for a quest with it later. Chyos just happened to see patrons mentioning that there aren't any pets, so he put one in for fun.
I've noticed that if Kate's friendship level is too high you can't corrupt her with the cumwine. Another issue I found that when you try and have a drink with your friend, the MC has an inner dialogue about Kate being ok and needing to listen to her convo with the inn keeper. Sorry I can't remember all the names, but this game is making me mad right now. I don't know how many times I've had to restart or go back quite a few saves because the coding in the game seems pretty screwed up with the different variables that trigger certain events. Don't even get me started on the whole "favor" system with Penny. If the dev wants you to get to know her why would the MC be punished for talking to her or sleeping in her house?
Being unable to corrupt with the cumwine shouldn't be possible. However, there is a part where you should be corrupting her before giving her cumwine. It is when she joins the MC and Rick at their table. With enough affection, Kate will take a sip of the MC's drink, and MC has the option of sneaking cum into the drink before she drinks it. This will cause her to get corrupted. I haven't heard of this needing to be done to a certain point before cumwine will work in her bedroom, but I suppose you could try it to see if that makes the difference. Please let us know if it does. If it does not, I expect that you're using a mod or the in-game cheatbook. Note: If you don't know how to get her to sit down, it is during the serving game, you have to walk her to the MC's side of the table to start the conversation, and eventually she will sit down once she has enough affection.
The inner dialogue thing is about making sure to be on the second floor of the inn at midnight, so the MC can witness the conversation between Kate and Thomas (the innkeeper). It happens if she doesn't do a good enough job serving the guests during the serving game earlier in the night.
What is the "favor" system with Penny that you mention? I do not understand this. The only punishment I can think of for talking to her is when she is praying in her parents' bedroom before you've developed enough affection with her from dining with her at the table at dinner time. I don't know anything about being punished for sleeping in her house. I haven't played through NTR route with her getting back with Henry though, so maybe there is something with that? Not sure.
Quest Arenfield militia. After named mercenary group, next goal - sit with lyvia at camp in the evening, nothing happens.
Do you have wine? Try also having Sleepwine in your inventory, I've heard of a bug with that, although it should be fixed in latest version so that you shouldn't need Sleepwine.