Should I remove the foam when cooking chickpeas? Whenever I boil chickpeas aka garbanzo beans, I usually scoop up and discard the foam that rises to the top of the pot. Is there any reason other than for aesthetics to remove the foam?  

As a secondary question, has anyone tried using this foam as an ingredient? It seems like it contains a lot of protein because the bubbles are stiff and resistant to popping. The flavor is pleasant, and the texture is unique.
According to a quick search it appears that your notion about it being protein-based is correct.  Most of the recipes I've seen say to skim it; the above-linked site says that adding a little oil will keep the foam down.

I personally wouldn't do anything with it as an ingredient unless I had a truly massive amount of it to experiment with--I don't know enough about the properties to make anything other than wild guesses about how it would work.  The only similar material I can think of would be beaten egg whites, but unless you were desperate for a vegan alternative and were already boiling up vats full of chickpeas I'd just use the eggwhites.