While I don't know the flat-Earth answer to the sails question (which I always considered a particularly damning bit, especially when you consider that this happens regardless of where in the ocean you are or where the other ship is coming from), I can assure you that on a flat Earth, the horizon would still appear round. In fact, on a round Earth as well as a sufficiently large flat Earth, in reality, the horizon appears to be a straight line. This is because you are only looking at a small part of it: the rest of it wraps around behind you, in a circle. The reason the horizon exists at all is a perspective effect, a somewhat lengthy issue in geometry that more or less says, in a perspective projection (e.g. what your eyes do), parallel lines appear to converge, and the locus of the intersections of all parallel lines is the horizon.
As far as the circumnavigation issue, I suppose a diehard flat-Earther would assert that such claims are lies or lunacy.
-Erasmus