Have you considered the possibility that your cooling coils (evaporator) may need to be cleaned? None of us mentioned it but as the coils become coated with dirt the efficiency of them will decrease and over time it can dramatically lower the ability of the a/c to cool the inside air. That same thing is true for the condenser coils if they should become plugged with foreign materials such as cottonwood seeds or most anything else. The inside tends to get bad first due to it normally being damp to wet, but outside can also be plugged with flying debris.
No. The blower of the roof air conditioner gets power from the same supply as the compressor and all other parts. In addition, the blower uses an alternating current motor and those cannot be operated from direct current as the design of each is quite different.
It could be run via an inverter if your a/c unit has a "fan only" selection, as most do.
In my opinion, it isn't practical to do this, but it is possible. As to the cooking grease issue, that would be no different than what happens when you use the a/c while cooking. The cooling coils do get a buildup of dirt and grime over time and should be cleaned occasionally. Most of them do not get cleaned until problems develop so the result is that the a/c becomes less efficient as it gets dirtier.