The answers at this related question might be of interest.
As implied by the comments, there is a large body of work on this topic.
Here are some authors (definitely not exhaustive) who have worked out the exact structure of character varieties of 3-manifold groups:
In particular, the answer to your first question is yes. See here for torus knots and $n=3$ and here for the figure eight knot and $n=3$yes. See:
- Geometry of the SL(3,C)-character variety of torus knots, by Vicente Muñoz, Joan Porti for torus knots and $n=3$, and
- The SL(3,C)-character variety of the figure eight knot, by Michael Heusener, Vicente Munoz, Joan Porti for the figure eight knot and $n=3$.
For your second question, I recommend reading generalities about tangent spaces to character varieties in:
hereCharacter Varieties, by Adam Sikora for generalities. With
With respect to local deformations for (finite volume hyperbolic) 3-manifold groups, this and thisthe following references answers your second question positively.:
- Local coordinates for SL(n,C) character varieties of finite volume hyperbolic 3-manifolds, by Pere Menal-Ferrer, Joan Porti and,
- Twisted cohomology for hyperbolic three manifolds, by Pere Menal-Ferrer Joan Porti
As to the third question, I am not sure what "remarkable" means here, so I will just leave that one alone.
Another interesting part of the story of character varieties of 3-manifold groups concerns dynamics. See the very nice exposition by Dick Canary titled hereDynamics on character varieties: a survey (and references therein).
