Hi Amy! First of all, yes, this is a very common expression. In English, it is also common to say, "down to the very last drop" when talking about drinking something. Number 1 is correct! It does not mean he is eating like a mole. Instead, it means that he ate everything, even the crumbs. In this case, "down" is used meaning "to the extent of". "Down to the very last morsel/crumb" or "Down to the very last drop" are just common expressions. I believe these are the only common instances when "down" is used this way. I hope this helps you.

It means he took it all in until the very last piece was finished. "Down to," is just an expression that basically means, "To the." Don't take it literally. If you take a chicken home and your family eats it all, you can say, "My family ate the chicken down to the very last bone (or morsel)."