What's the deal with half-walls behind the kitchen cabinets? Can I cut into it?
Aside from electrical and plumbing interference, you can do pretty much whatever you want with the framing. That wall is mostly decorative (it saved some money by not requiring the cabinet makers to finish the backs of the cabinets). 

If you have smooth walls, open the drywall enough to work. Frame in your opening, replace the drywall, add corner bead or wood trim, and line the opening however you had planned. 

If you want more specific help, strip out the kitchen's life story up there in your post, put up a photo of the actual project area, and we'll do our best.