The results proved in
S. Kojima, "Deformations of hyperbolic 3-cone-manifolds",
J. Differential Geom. 49 (1998), no. 3, 469-516
provide complete answers to questions 1 and 3.
The main theorem of the cited paper ensures that any compact hyperbolic cone metric with cone angles at most π can be continuously deformed to a complete hyperbolic metric on the complement of the singularity. Furthermore this deformation can be done by an angle decreasing deformation.
This provides a partial answer to your question (2) and a complete answer to (3): If $K$ is the singular locus of a cone structure with angles $\leq \pi$, then $K$ is necessarily hyperbolic. Moreover, thanks e.g. to a suitable version of the Schlafli formula, the hyperbolic volume of the cone structure is stricly smaller than the hyperbolic volume of the cusped manifold $S^3\setminus K$, which in turn is equal to the simplicial volume of $S^3\setminus K$ (up to the constant $v_3$, which I guess you forgot to mention when asking question 3).
Moreover, Theorem 1.2.1 of Kojima's paper implies that, if $S^3$ supports a cone structure with singularity $K$, then $K$ must be hyperbolic, regardless of the cone angle of the original cone structure along $K$.