7500 is used for hijacking, 7600 for radio failure, and 7700 for any other emergency. I understand that 7600 and 7500 need special handling (you can't talk to ATC if your radio has failed, and a hijacked plane may need a military escort), but why not for other emergencies? For example, why isn't there a "fire" code so the airport can have firefighting equipment ready?
The airport should have firefighting equipment ready in the event of any emergency. Whether there is already fire on board, a crash will start a fire about 90% of the time.
Every emergency is unique and different. Their handling by ATC is mostly the same: give the plane absolute priority, ground or divert whoever you have to, have equipment ready, be prepared to call for rescue. Not much more that can be done.