Existence of graphs of every order related to Barnette’s conjecture
I have a partial solution, with constructions for the first question and for the second question when $n = 2\ (\text{mod}\ 4)$.

Here are constructions for $n = 8$ and $n = 14$. These generalise to constructions for $n = 0\ (\text{mod}\ 4)$ and $n = 2\ (\text{mod}\ 4)$, by taking any square, for example 1,2,3,6 in the second construction, and replacing it with a ladder like shown in the third image.

For the second question, the $2\ (\text{mod}\ 4)$ construction has no nontrivial automorphisms when $n \geq 18$, as we can recover the labeling. First find the added ladder and the pairs 1-2 and 3-6. To discern between them, note that the nodes 1 and 2 are not in any four-node cycle except in the ladder, while 1 and 6 are. Similarly 2 and 3 are not in a four-node cycle except in the ladder, while 1 and 6 are. Then you can uniquely discern 4 (only non-ladder node connected to 1), 5 (same for 6), 8 (only node connected to 3 and 5) and so on.