Can I request a caterer to make specific dishes? Birthday cakes?

Generally it's easier to have baked the cake yourself and bring it to the party and light it there rather than have the caterer do it. And yes, if you hire a caterer for the birthday party they will eventually show up and eventually get to baking a cake on their own. I noticed this when I forgot to bring the cake to the party lot and the caterer made a cake themselves anyway (though I quick-baked a new cake on the lot regardless).

Oh, and I choose to keep party lots either on your own home lot or non-residential lots not your own to avoid being ousted if not having the Always Welcome trait. I learned this the hard way the first time I learned of this.