Do two empty sets have any elements in common?

Your argument looks good, $\varnothing\cap \varnothing=\varnothing$. In fact $\varnothing \cap X=\varnothing$ for any set $X$ !

You are right. In particular, $\emptyset$ is not a common element, but rather a common subset. That is: $\emptyset$ has no elements, but is indeed a subset of itself (and of every other set!).

I think that two empty sets do not have any element in common since they do not have any elements in the first place.

Exactly.

Should I count $\varnothing$ as a common element?

No, because $\varnothing$ is not an element of neither of the sets. It only is an element of their power set (i.e. a common subset), but the empty set is not an element of the empty set itself and therefore not a common element of the intersection between two empty sets.