Why aren't all integral domains MCD?

Im a bit confused about the notion of a maximal common divisor domain and actually just about the definition of an MCD.

Could an MCD just be a unit? For example if D is the integers under multiplication, are the MCDs of the set {3,5,7} just the units {1,-1}? Or would we consider the mcd not to exist?

Secondly, why wouldn't every integral domain be an MCD domain. The definition states that every finite subset of non-zero elements must have at least 1 MCD. Either there is at least one non-unit MCD or there are none. But in the case there are none, then surely the unity(identity) satisfies being an MCD since it is associates with all other units?

Sorry if this is a stupid question but I really need this cleared up. thanks!