This question (which is more a curiosity than a research problem) originates from these two:
The first question basically asks: does there exist a non-dense, open subset of $S^2$ whose boundary contains no image of injective paths? I think that this reduces to asking for an open, non-dense set whose boundary is totally path-disconnected.
If we remove the word "path", then the answer is given in question 2. Is the answer easy/known/unknown including the word "path"?