Minimal area of Seifert surfaces

In question (1), if you allow $g$ to vary, then this is answered positively by Hardt and Simon (see also).

The answer to question (2) is no. Almgren and Thurston construct unknots which do not bound an embedded disk in their convex hull. If $\mathscr{F}$ (fixing $g=0$) contained a minimal area member, then it would have to be a minimal surface. However, a minimal surface bounding $K$ must lie in the convex hull of $K$, a contradiction.

Yes, the surface is smooth, and you can get one of minimal genus. This follows from the references in the paper of Edmonds that you cite. The paper of Freedman-Hass-Scott shows that the least area surface in the homotopy class of a minimal genus Seifert surface is actually smooth. It is a useful observation that this minimization also works in the larger class of piecewise smooth surfaces. This comes up when (as in Edmonds's paper) you are doing cut/paste arguments with a pair of surfaces.