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Here is an attempt at a 'definitive summary'. To begin with positive results: $\mathsf{CH}$ implies a “yes” answer to this question. The fastest way to see this is to first embed a given partial order …

No, $\mathrm{RO}(X)$ is complete; $\mathcal{P}(\omega)/\mathit{fin}$ is not (no strictly increasing sequence has a supremum).

Here is a provisional negative answer. If $\mathcal{C}$ is a c-minimal chain then $N=\bigcap\mathcal{C}$ is closed and nowhere dense (if nonempty): if the interior is nonempty take a nonempty regular …

Look at your and my second answer to the original question. Take your map from $\mathcal{P}(\omega)$ into $\mathcal{L}$ (or rather the set of functions before identifying almost equal elements). That …

Here are a few tricks to play with. For every $A\subseteq\omega$ define $f_A$ by $f_A(0)=0$, $f_A(n+1)=f_A(n)+1$ if $n\in A$, and $f_A(n+1)=f_A(n)$ if $n\notin A$. If $A\subset^*B$ (so $B\setminus A$ …