On Bipartite planar graph (again)
If $\min(m,n)=1$, the maximum number of edges in a planar bipartite graph with $m$ vertices on one side of the bipartition and $n$ on the other side is $m+n-1$ (just connect them in a star). Otherwise, the maximum number is $2(m+n)-4$.

The upper bound (any bipartite planar graph with $V$ vertices has at most $2V-4$ edges) follows by combining Euler's formula $V-E+F=2$ with the observation that all faces have at least four edges, and each edge belongs to at most two faces, so $E\ge 2F$. Use this inequality to eliminate $F$ from Euler's formula and you get $E\le 2V-4$.

For the lower bound, choose two vertices from the side with $m$ vertices and form a complete bipartite graph $K_{2,n}$ with $2n$ edges. Then, choose one of the quadrilateral faces of this graph and place the remaining $m-2$ vertices into it, connecting each one to both of the vertices of the other color within the same face, adding $2(m-2)$ more edges.

As for the number of distinct maximal planar bipartite graphs on this many vertices, I don't know, but it's certainly some kind of exponential.