This is a fairly sorry counterexample to the main conjecture, but I do think it is one. Take $n=2$, $F_{0}=\{0\}$, $F_{1}=\{1\}$. Then neither $0$ or $1$ satisfy the conditions you're looking for on $l$.

This is a fairly sorry counterexample to the main conjecture, but I do think it is one. Take $n=2$, $F_{0}=\{0\}$, $F_{1}=\{1\}$. Then neither $0$ or $1$ satisfy the conditions you're looking for on $l$.

Similarly, for Question 2, in a two-element anti-chain with $x\leq x$, $y\leq y$, but no relation between the two, there's no such $h$.