Short answer
Yes. Both questions have already been settled in the affirmative.
For an orientable Haken 3-manifold one can algorithmically produce a finite presentation of its mapping–class group, and from this decide

(a) whether the manifold possesses an orientation–reversing self-homeomorphism, and

(b) whether two given orientable Haken manifolds are homeomorphic by an orientation–preserving map.

Where to find it
The two standard places where the oriented problem is written out are

1. G. Hemion
   “The Classification of Knots and 3–Dimensional Spaces”,
   Oxford Sci. Publ., 1992.
   Chapter 13 (especially Theorem 13.4.10) gives an explicit algorithm that, for any two triangulated orientable Haken 3-manifolds, decides whether there is an orientation–preserving homeomorphism between them. In particular, applied to a single manifold it decides whether the orientation can be reversed.

2. S. Matveev
   “Algorithmic Topology and Classification of 3–Manifolds”,
   Springer, 2003.
   Section 1.4 (“The mapping-class group of a Haken manifold is algorithmically computable”) and the beginning of Chapter 7 give a modern version. Theorem 1.4.14 together with Corollary 1.4.16 states:

   “There is an algorithm that, for a triangulated Haken 3-manifold M, constructs a finite presentation of the group Homeo(M)/isotopy. Consequently one can decide in particular whether Homeo(M) contains an orientation–reversing element, and whether two orientable Haken manifolds are homeomorphic through an orientation-preserving map.”

How the algorithm works (outline)
The main ingredients that make the procedure effective already appear in Haken’s original programme; what has to be added is explicit control of the mapping-class group.

1. Canonical JSJ decomposition.
   Jaco–Shalen/Johannson give the structure theorem; Jaco–Oertel and Jaco–Tollefson supply an algorithm to carry it out from a triangulation.

2. Pieces of the decomposition.
   • Seifert–fibred pieces: the Seifert invariants are computable and tell when a fibre-preserving homeomorphism reverses orientation.
   • Hyperbolic pieces: the Epstein–Penner canonical cell‐decomposition or Weeks–Manning canonical triangulation is algorithmic; it gives the full (orientation-preserving and reversing) isometry group.

3. Gluing data.
   Johannson shows that, up to isotopy, any self-homeomorphism of a Haken 3-manifold is obtained by: – a symmetry of the JSJ graph,
   – for each piece a homeomorphism as in (2),
   – finitely many Dehn twists about the JSJ tori.
   Jaco–Tollefson give an algorithm to generate these twists. Hence a finite list of generators for the whole mapping-class group is produced effectively.

4. Deciding orientation questions.
   The algorithm labels each generator by the sign with which it acts on the fundamental class in H₃; linear algebra over ℤ detects the presence (or absence) of an element of sign –1, and, for two manifolds M and N, tests whether every homeomorphism determined in steps (2)–(3) preserves orientation.

Because every step is effective, Questions 1 and 2 have algorithmic solutions already recorded in the references above.

Thus the oriented homeomorphism problem for Haken 3-manifolds is not open; it was settled in the literature more than two decades ago.