Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a homeomorphism $f: M \to M$ with $f(K_1) = K_2$?

I would expect these questions to be involved (at least as difficult as the recognition problem for 3-manifolds) but I thought that the answer is possibly a folklore consequence of geometrization. The special case of the first question for $K_1$ unknotted follows from a result of Hass and Lagarias (see Theorem 1.2). The case where $M = S^3$ is a result of Waldhausen using normal surface theory.

As an aside, I don't know anything about the corresponding questions where instead of considering knots we consider closed embedded surfaces $F_1$ and $F_2$. Even in the case where $M = S^3$ (so only the isotopy question is relevant) the genus of $F_1$ and $F_2$ is greater than one, this is a total mystery to me. By looking at tori bounding regular neighborhoods of knots, this is strictly more difficult than the corresponding questions for knots. Are there results in this direction?