I really think you must upgrade your skills on cooking first, it won't be temporary unlocked for birthdays or dinner-parties. About 'tell the birthday sim a joke", do you have a wife/husband or kids who you can let that do? I never saw that message in my 'to-do list' on birthdays because i didn't had a marriage at that time.
When you scroll down this page you can find all unlocked foods per level.
PS. Make sure that once you actually did bake a cake you need to add candels on them manually. I did had a 'mission' to blow the candles. So in case you get that in the future, click on cake and add them.
Anyway, the first cake i see in that list (glanced over it) is unlocked at level 5 (signature cake) and on 6 you will have confetti-cake.
On topic on the other issue, weird indeed that you get the thing to tell the birthday sim a joke. Did you hire yourself as entertainer? (although i don't think that's possible because it then switches you from having it's birthday to the entertainer).
