### Finding the order dimension of a set of lattices

Let $L_n$ denote the set of all lattices with $n$ points in SAGE.
I want to find for given $n$ (lets say $n$ is smaller than 10 or so) all lattices with order dimension at least 3. For the definition of order dimension see https://www.encyclopediaofmath.org/index.php/Dimension_of_a_partially_ordered_set.
I can do it for a concrete example, but I am not sure how to obtain a (graphical if possible) output of the result for a given $n$. Here the concrete example for the boolean lattice with 3 elements:

P=posets.BooleanLattice(3)
dimension(P)

I also have two small mathematical questions:

I hope I see it correctly that the order dimension of a lattice is 1 if and only if this lattice is a chain?

Are there tables on the internet or the literature on known results of what the order dimension is for lattices coming from combinatorics such as the divisor lattice. Do I see it correcty that the order dimension of the divisor lattice of a natural number n is equal to the number of prime divisors of n?