Is the empty set a subset of itself? Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty set, have infitinite emptiness e.g. do all sets including the empty set contain infinitely many empty sets?
There is only one empty set.  It is a subset of every set, including itself.  Each set only includes it once as a subset, not an infinite number of times.
An empty subset can be selected from any set, including the empty set itself, using selection criteria that cannot be satisfied, e.g. selecting those elements $x$ such that $x\neq x$.
See my formal proof (in DC Proof format) at: http://dcproof.com/ExistenseOfNullSet.htm
There is only one empty set. If $\phi_1$ and $\phi_2$ are empty sets, then $\ \phi_1 = \phi_2$.
See my formal proof (in DC Proof format) at: http://dcproof.com/UniquenessOfNullSet.htm

Edit: For any set $X$ we can select an empty subset $S$ such that:
$\forall a:[a\in S\iff a\in X\land a\neq a]$
or
$\forall a:[a\in S\iff a\in X\land a\notin X]$