What can I do about this gap between the sill/exterior board and the subfloor?
There should be no gap.  Looks like shoddy construction.

See figure 16 from this page for proper alignment:



Also I have never seen a "band" or "rim" joist that are configured in the fashion described by your image.  It can't be good that half of the wall is resting out in space.

You could rip a board to fit the gap so that the weight of the wall is transferred to the foundation evenly, but I would probably call a professional to see if that option is even structurally sound.  I would not fill it with spray foam.  Fix the structure first, then insulate.