Hmmmm, for the "cat care" type jobs, I could see there being a few variations of cat care missions, where they all go something like: 1. Be told what cat you need to track down and perhaps the general direction (somewhat like dead letter missions), 2. Use tracking/guesswork to find the cat, 3. Capture the cat--preferably by luring it in with a treat or toy, but by sheer grappling if necessary, 4. (Varies based on mission.)
Like medicating it could mean feeding it a pill that you hide in a treat (co.hi.obj). Or wrapping it in a towel to give it liquid medicine. Or there could be a cat that's gotten into a very dusty attic and needs a bath. Or that has burrs in its fur and needs to have them brushed out.
There could also be simpler, low-level cat missions where you just need to fill some water or food bowls, repair catnip toys, or play with some bored cats.
Maybe cat-related missions could be based out of the cat shrine?
PS I especially like the idea of filling spiced coffee orders as a mission. :) Maybe there could even be a harder one where they don't tell you outright what spice they want, but they say they want "something warming" for example and you have to see what spices fit the bill. You could give them the coffee and they could tell you whether they like it (giving more hints in the process). "Hmmm... interesting, but I don't like the woody flavour." or "Not bad, but I think a nuttier flavour would be nice, too." So the intent would be that you'd have to try a few to narrow it down, unless you got lucky.