For k = 1, we can solve in polynomial time via brute-force search. If I assume that my orthogonal convex polygon takes up at most $n$ horizontal units and $n$ vertical units, then a single rectangle is defined by an interval on the x axis and an interval on the y axis. There are $O(n^2)$ such intervals on the x axis and $O(n^2)$ on the y axis, for a total $O(n^4)$ runtime. I'm pretty sure the exponent can be reduced by being a little smarter with pruning intervals.

Yes, the rectangles should be inscribed in the polygon. Every edge of the polygon and every edge of the k rectangles should be either horizontal or vertical.