Let $P_n$ denote the poset with elements $P_n=\{1,...,n\}$ ordered by divisibility and let $L_n$ denote the distributive lattice of order ideals of $P_n$, whose elements should correspond to primitive subsequences of $\{1,...,n\}$, see https://oeis.org/A051026.

>Question 1: Is $L_n$ Sperner? What is the width of $L_n$?

The width sequence start for $n \geq 2$ with 1,2,2,4,4,8,10,15,21,40,45,87 and the poset is indeed Sperner for $n \leq 15$.

>Question 2: Let $F_n$ be the free distribitive lattice given as the set of order ideals of the Boolean lattice $B_n$. For which $n$ is $F_n$ Sperner? It is Sperner for $n \leq 4$. What is the sequence $a_n$ of width of $F_n$? The sequence start with 1,2,4,24 for $n \geq  1$.

More generally one can ask this question also for the distributive lattice of order ideals of the divisor lattice of $n$ (all divisors of $n$ ordered by divisibility), see the comments.

I would be very interested to see what $a_5$ and maybe $a_6$ are and see if this sequence appears in the oeis.