Is the empty set the only possible set for $A$ such that $A=\{x|x\not\in A\}$? Is the empty set the only possible set for $A$ such that $A=\{x|x\not\in A\}$?

Not even the empty set has that property.
For any set $A$ we can show $A\ne\{x\mid x\notin A\}$. Namely, either $42\in A$ (in which case $42\notin \{x\mid x\notin A\}$), or $42\notin A$ (in which case $42\in \{x\mid x\notin A\}$). In both cases we have found something that is a member of exactly one of the collections, so they're not the same.
