1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. If this were false, Vassiliev invariants would not separate knots by a result of Kuperberg: _Detecting knot invertibility_, J. Knot Theory Ramifications 5 (1996), 173-181 arXiv:[q-alg/9712048](https://arxiv.org/abs/q-alg/9712048). This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), _[Vanishing of 3-Loop Jacobi Diagrams of Odd Degree](https://arxiv.org/abs/math/0511602)_ (J. Combin. Theory Ser. A. 114 (2007) 919-931 https://doi.org/10.1016/j.jcta.2006.10.005) where we prove this for 3-loop Jacobi diagrams. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links (see the paper _[Vassiliev homotopy string link invariants](http://www.math.toronto.edu/~drorbn/papers/homotopy/homotopy.pdf)_ by Bar-Natan (Journal of Knot Theory and its Ramifications 4-1 (1995) 13–32. https://doi.org/10.1142/S021821659500003X).
1. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
1. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is the survey paper _[The Fundamental Theorem of Vassiliev Invariants](https://arxiv.org/abs/q-alg/9702009)_ by Bar-Natan and Stoimenow.