If I have this correct the card that the airport plugs into is an I/O card, I think it has all the external ports on it including the magsafe (ties in with may crash faster when on mains). Therefore the hardware chain is airport card into I/O card into logic board. Before spending anymore money I would try the following:
-Reset PRAM - power on and then hold command-option-P-R wait until you have heard the boot chime a couple of times, it would also do not harm to reset the SMC pop battery out and unplug from mains and then hold power button down for 10 seconds, I am not sure if this ties in with airport or not but as magsafe and airport card are on the same board I am guessing.
-Retest to see if problem still present.
-Reinstall OS and apply all updates, at this point I would not install any other apps, not clear from above if you did or not.
-Retest.
-Next I would open up the unit and disconnect and reconnect all the cables relating to the hardware, so that would be the the airport card into the slot and the connector between the I/O board and the logic board. I also get a can of compressed air and blow all connectors just to be on safe side.
-put it all back together and you guessed, retest.
If you suspect heat download a monitor app to see if it shows the machine getting warm, I like iStat for this sort of thing. The other thing is to try and note times when you start and times of failure to see if it is the same length or varying. It will help with knowing if you have fixed it.
If none of the above work replace hardware, starting with airport card (which as long as it was from a reputable source should be OK so not really the next thing as already done), so then the I/O board, if that does not work you are looking at a logic board which is expensive.