Quick question about well-orders:

Or put differently:

Is every set in bijection with a well-ordered set well-ordered?

Yes.

Let $S$ be in bijection with a well odered set $W$ with $f: S \to W$. Define a well-order on $S$ as $s_1 \leq s_2 \iff f(s_1) \leq f(s_2)$.

Then show that this defines a well-order on $S$.