Let $\kappa$ be a regular ordinal.
Say that for $f,g: \kappa \rightarrow \kappa$, $f \leq g$ iff $f(\alpha) \leq g(\alpha)$ for sufficiently large $\alpha$. Now define $(X_{\kappa},\preceq)$ as the quotient poset $({\kappa}^{\kappa},\leq) / \sim$ where $f \sim g$ iff $ f \leq g \wedge g \leq f$.
Note that $X_{\kappa}$ is never well-founded since $\mathbb{N} \ni n \longmapsto (\kappa \ni \alpha \mapsto \alpha - n)$, it being understood that if $\alpha$ is not the $n^{th}$ successor of some ordinal, then $\alpha-n$ is the biggest limit ordinal $\leq \alpha$, is stricly decreasing. (or rather the sequence of equivalence classes is)
I have three questions regarding this relatively simple construction, possibly in decreasing order of difficulty:
$(i)$: Do we know the supremum of the set of ordinals which embed in $(X_{\kappa},\prec)$?
$(ii)$: Do we know $cof(X_{\kappa},\prec)$?
$(iii)$: Is there a definable embedding of $\omega_1$ into $X_{\omega_0}$?
It isn't difficult to see that $cof(X_{\kappa},\prec) > \kappa$ because given an increasing sequence $u: \kappa \rightarrow X_{\kappa}$, the equivalence class of $f:\kappa \ni \alpha \longmapsto \sup(\{u(\beta)(\alpha) \ | \ \beta \leq \alpha\}) + 1$ is a strict upper bound of $u$. However I can't think of a constructive way to embed (cofinally or not) ${\kappa}^+$ in $X_{\kappa}$.