Can a "crippled" aircraft carrier launch airplanes in Midway?
I don't think this idea is a valid simulation. The two most critical tasks of an aircraft carrier, in order, are staying afloat and launching/retrieving it's planes. Provided the crew is not fully dedicated to staying afloat (ie there remains one or more hit points), the crew will continue to launch and retrieve plans.

Remember the reaching of zero hit points may not be a simulation of actually having sunk - but rather of having become unoperational as a carrier.

Of course these classic TAHGC games simulate in very broad brush strokes - it is common now to improve the simulation with more complex rules. I have not played Midway specifically, but at the cost of additional complexity I would consider slowing a carrier's launching and retrieval of planes once it had only one or two hit points left. This would simulate the difficulties of continuing operations a damage built up. However a carrier that cannot launch and retrieve at all is a sitting duck. Somehow, someway, some planes will be launched until the crew is forced to abandon ship. 

Additionally, as a simulation of the different national characters and cultures of Japan and America, I would constrain damaged American carriers to only launch planes that could reasonably be expected to be retrievable - but make this optional for the Japanese captains.  An early kami kaze option if you will.