For question 1) Yes analogs of knot invariants exist for any manifold regardless of prime decomposition or Thurston decomposition. For example, consider the Alexander polynomial and its categorification, knot Floer homology. These can both defined for general one cusped 3-manifolds. Computations of these invariants have been used lately to answer the question: Given a homology class $\mathcal{s}$ in a manifold $M$, is there a knot $K$ such that as a curve in $M$, $K \in \mathcal{s}$ and $S^3-K' \cong M-K$ for some knot $K'$ in $S^3$. This question is equivalent to if $M$ can be realized as surgery along a knot in $S^3$.
For example, Josh Greene was able to show here that if a knot in $S^3$ admits a lens space surgery then its knot Floer homology appears on a confined (but infinite) list of known examples, here. Margaret Doig offers a nearly complete classification knots in elliptic manifolds that admit an $S^3$ surgery in these two papers. And recently, Yi Ni and Xingru Zhang places significant restrictions on knots in certain Nil manifolds that could admit $S^3$ surgeries here. These are all examples of exploiting knot invariants in closed manifolds.
For question 2) I think the consensus is that knot invariants are more delicate in non-simply connected manifolds. As brought up in the comments, there could be several different analogs of the unknot. One analog of unknottedness is minimal with respect to Seifert genus, i.e. a knot is minimal if for all simple closed curves in its homology class of a manifold there is no curve that bounds a lower genus (orientable) surface. However, even this question is considerably harder. There is some hope that there might be a polynomial time algorithm for recognition of the unknot in $S^3$. Along these lines, Agol, Haas, and Thurston show that just determining the genus of a knot in general 3-manifold is NP-Hard, here.
