Chow mein is a stir fry of noodles. It is a corruption of the Chinese “chao mian”. When someone says “Let’s chow down” does this expression come from this chow? Possibly, but nothing is conclusive.
Chow meaning food in American slang perhaps originates with the Chinese railroad workers. All we know is that it comes from Chinese pidgin English chow-chow a reduplication of the Chinese word cha meaning mixed.
The dog chow-chow probably comes from the Zhou Dynasty and is not related to food.