Do two empty sets have any elements in common? I think that two empty sets do not have any element in common since they do not have any elements in the first place. 
Should I count $\emptyset$ as a common element?
Edited: two empty sets as: $A=\{\;\}$, $B=\{\;\}$.
 A: Your argument looks good, $\varnothing\cap \varnothing=\varnothing$. In fact $\varnothing \cap X=\varnothing$ for any set $X$ !
A: 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!).
A: 
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.
A: $A$ and $B$ do not have any elements in common.
$$
  \not\exists x: x\in A \wedge x\in B
$$
However, $A$ does have all its elements in common with $B$ and vice versa!
$$
  \forall x \in A: x\in B
$$
This is of course a vacuous truth.
