As far as I know, question 1 is open.
Let me say a few words about the first part of question 2. One way to associate a q-series to a 3-manifold is via the 3-d index developed by Dimofte, Gaiotto and Gukov:
Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17, No. 5, 975-1076 (2013). ZBL1297.81149.
Dimofte, Tudor; Gaiotto, Davide; Gukov, Sergei, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325, No. 2, 367-419 (2014). ZBL1292.57012.
To concretely associate this q-series to an invariant of a 3-manifold, Garoufalidis, Hodgson, Rubinstein, and Segerman were able to show that for any cusped orientable hyperbolic 3-manifold there is a standard 3-d index which can be computed from its canonical cell decomposition. The proof of their main theorem is worth flushing out here. They show that for any triangulation of cusped 3-manifold which is 1-efficient (a property involving normal surfaces) the 3-d index makes sense to compute. Furthermore, if two 1-efficient triangulations are connected by a single 2-3 or 3-2 move (plus 0-2 and 2-0 moves) the 3-d index is the same for both triangulations and if a manifold does not admit a canonical triangulation then all simple refinements of its canonical cell decomposition are 1-efficient and related by simple moves as above. Finally, their paper also includes some computations and shows that the 3-d index of a triangulation converges as a formal q-series (that is it can be written down) if and only if the triangulation is 1-efficient.
Garoufalidis, Stavros; Hodgson, Craig D.; Rubinstein, J.Hyam; Segerman, Henry, 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, Geom. Topol. 19, No. 5, 2619-2689 (2015). ZBL1330.57029.
Garoufalidis, Hodgson, Rubinstein, and I did a follow up to this paper, which tries to better understand the connection between the 1-efficiency and convergence of the 3-d index:
https://arxiv.org/pdf/1604.02688.pdf
An interesting question along these lines, is what is the 3-d index counting? There is some hope that trying to 'categorify' this invariant might shed light on that question as well. In my opinion, this is a relevant interesting question which has already motivated novel approaches to better understanding 3-manifolds.