How do I agree to: I'm in no mood to do the dishes. - Neither am I. (meaning I'm not in the mood ...) I'm in no mood to do the dishes. - So am I. (positive/not negated verb "am" hence So...)?
Formally, one would use "neither am I." I am not 100% sure about why this is as you're right that it's not strictly phrased that way -- at least where I'm living one would say "I am not in the mood to do the dishes," and the response makes sense there. I would say that in general "so am I" suggests that you misheard them as saying "I'm in the mood to do the dishes" and were suggesting that you both do it together; the redundancy of "neither" helps to clarify that you understood them properly. Other common but less-formal responses are "me too" and "me neither," and "me too" definitely has the same confusion associated with it: you would almost certainly say "me neither."
Informally, another common response in my location is "same here," which does not suffer this ambiguity.