What is the best way to install a bathroom exhaust fan in a short, dropped ceiling? From what I can tell without having removed anything, the tiles look to be at most 3 inches below the actual ceiling underneath. Are there fans which include a housing that can expand downward to fill the few inch void between the grill mounted on the tiles and the actual ceiling? If not, then I assume some type of duct work needs to be added? I have not yet purchased the fan.
There's really no extension involved. You'd suspend the fan flush with the drop ceiling tile as you normally would with drywall. The specific hardware used depends on the fan. You could DIY by simply adding metal or wooden crossbars with sheet metal screws to the fan housing that rest on the tile and extend over the grid rails. Some kits may offer bars similar to what you'd get with a recessed light canister.
You'd also need to create clearance in the drywall ceiling. If there's no insulation there, that's easy. Just give yourself enough room to work and hack out a rectangle.
If you're in an insulated ceiling you'll need to box it out and be sure to replace the insulation above, along with any vapor barrier present.