Assuming countable choice, $\omega_1$ is first countable and countably compact but is not separable or Lindelöf

Assume countable choice. Successor ordinals and $0$ are isolated. If $\alpha<\omega_1$ is a nonzero limit, enumerate the at most countable ordinal $\alpha$ and take the successive finite suprema of that enumeration; the result is a countable cofinal sequence, and the corresponding final intervals $(\beta_n,\alpha]$ form a local base at $\alpha$. Thus $\omega_1$ is first countable. The published ordinal theorem makes it countably compact.

Every at most countable subset $D\subseteq\omega_1$ is bounded by some $\beta<\omega_1$, so the nonempty open tail above $\beta$ misses $D$; hence $\omega_1$ is not separable. The open initial segments $\{\,[0,\beta] : \beta<\omega_1\,\}$ cover $\omega_1$, but any at most countable subfamily has bounded union and therefore fails to cover. Thus $\omega_1$ is not Lindelöf.