at a minimum you will need a new brush dedicated to edible products (you'll never get 100% of your other paints out of your existing one)
You MAY be able to use your existing compressor IF you can dial the pressure WAY down! Standard pressures for most compressors will blow a hole in most icings!
A lot of us use the Airmaster from KopyKakes, it's specifically made for cake decorating!
I have a Paasche airbrush with a Scorpion compresor - bought at Hobby Lobby with their 40% off coupons. I have no problem with it blowing holes in icing.
Airbrushes can be taken totally apart. Maybe you can get it clean enough by taking it down to the parts and soaking everything in alcohol. I would be worried about just blowing cleaning fluid through it and thinking that would do it, though....
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