"Hence a set containing contradictions for all elements must be empty"
The empty set doesn't contain any element, so it doesn't contain propositions that are contradictions.
But it is true that $\forall x \in \varnothing : P(x) \land Q(x)$. That is, $\forall x: x\in \varnothing \rightarrow (P(x) \land \lnot P(x)).$ Since there is no $x \in \varnothing$, $x\in \varnothing \rightarrow (P(x) \land \lnot P(x))$ is vacuously true for any (every) such (nonexisting) $x$.
Perhaps what you mean to be saying is:
- $\varnothing = \{x \mid P(x) \land \neg P(x)\}:\;\;$ "The set of all $x$ such that $P(x) \land \lnot P(x)$."
Since there is no $x$ such that $P(x) \land \lnot P(x)$, the empty set remains empty.
The empty set can be defined by any condition(s) that no element can satisfy.
Examples:
Note: every universal statement about the elements of the empty set is true; this is known as vacuous truth. One might say that universal statements are "true until proven false." Since there are NO elements in the empty set, such statements cannot be proven false (as there is no element to serve as a counterexample).