We just bought a house and have repainted all the trim white to brighten up the house and modernize it. I love it. The only problem is that I would like for the kitchen to be more rustic, with an over-all red theme (red painted walls), dark countertops and backsplash, and brushed steel appliances. To fit this design, I'd like for the cabinets to be a darker finish, but the kitchen connects with the living room, which has white trim. Is it okay to paint the trim around the kitchen doorway white and not paint the trim in the kitchen white?

Absolutely it is okay. If the different color trim helps to define that space - go for it.
don't ever be afraid of different colors. Dif. color trim on door ways and windows can help define a new space within a large open floor plan. The real key, and I don't have a photo of your rooms, is to make sure the color break makes sense in defining the space.