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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:

1. Michael Heusener
2. Emily Landes
3. Melissa Macasieb
4. Vicente Muñoz
5. Kate Petersen
6. Joan Porti

In particular, the answer to your first question is yes. See:

1. Geometry of the SL(3,C)-character variety of torus knots, by Vicente Muñoz, Joan Porti for torus knots and $n=3$, and
2. 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:

Character Varieties, by Adam Sikora.

With respect to local deformations for (finite volume hyperbolic) 3-manifold groups, the following references answers your second question positively:

1. Local coordinates for SL(n,C) character varieties of finite volume hyperbolic 3-manifolds, by Pere Menal-Ferrer, Joan Porti and,
2. 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 Dynamics on character varieties: a survey (and references therein).