jsbaker
described what you have perfectly. And, yes, they are a pain in the ass to work on. Not only is everything above your head, try working on one straddling a bathtub and a bathtub with shower doors no less. One condominium complex I worked on that had blower coils above the bathtub had suspended ceilings with fiberglass ceiling panels in the bathrooms. The panels were held down by rocks, pieces of 2X4, bricks and anything else heavy enough to keep the panels from being sucked up by the blower because the units were uncased meaning the space above the ceiling and below the floor above was being used as the return air plenum. Being that you're seeing dust probably means your fan coil is uncased, too.
I would definitely have the system inspected and have something done about the return air in the kitchen. One other thing, I hope you're not concerned with efficiency because chances are regardless of the efficiency of the outdoor unit, you'll never get the efficiency it's rated at because of the limitations of the fan coil. Plus, there's a good chance the heat in that fan coil is electric. If you get the system inspected, which I urge you to do, make sure they check the system's operation, meaning they put manifold gauges on the condensing unit, paying particular attention to its operation (subcooling if the fan coil has an expansion valve or superheat if the fan coil has a fixed metering device). Those types of systems are notorious for inefficient operation and insufficient cooling due to their crappy design and installation. In most cases, the duct work is fabricated from fiberglass duct board material, which, if not fabricated and installed properly, has air leaks galore and is difficult, if not impossible, to access and seal because of it being installed above the ceilings.