Julia set containing smooth curve

I am not surprised that you found nothing in the literature.

On question 1: it has been studied under a much stronger assumption that the Julia set contains a smooth INVARIANT curve. Under some additional conditions, it was proved that such a curve must be algebraic. And there are such algebraic curves, other than circles. But without additional conditions this is not true. Nothing special can be said about Julia sets in these cases.

Invariant curves and semiconjugacies of rational functions

Circles and rational functions

The case when the Julia set is CONTAINED in a smooth invariant curve was studied in this paper.