Hardness of UNAMBIGUOUS-3DM
The answer to both questions is yes.
The standard reduction from 3-SAT to 3DM – as used by Garey and Johnson, for example – is not parsimonious, but there is a sequence of parsimonious reductions that goes:
3-SAT → 1-IN-3-SAT → MONOTONE-1-IN-3-SAT → 3DM
Moreover, these reductions can be done in a way that preserves planarity. The details can be found in [1, 2]. The key reduction MONOTONE-1-IN-3-SAT → 3DM is from [1]. It is shown to be parsimonious by [2], which also shows how to do the other reductions in a parsimonious way. The reduction from 3-SAT to PLANAR-3-SAT is done using the crossover gadget from [3], which [2] also shows to be parsimonious.
[1] M.E Dyer, A.M Frieze, Planar 3DM is NP-complete, Journal of Algorithms, Volume 7, Issue 2 (June 1986), Pages 174-184. PDF
[2] Harry B. Hunt, III, Madhav V. Marathe, Venkatesh Radhakrishnan, and Richard E. Stearns. The Complexity of Planar Counting Problems. SIAM J. Comput. 27, 4 (August 1998), 1142-1167. arXiv:cs/9809017
[3] David Lichtenstein. Planar Formulae and Their Uses, SIAM Journal on Computing 11:2 (1982), 329-343. doi:10.1137/0211025