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 $n=3$.
For your second question, I recommend reading about tangent spaces to character varieties here for generalities. With respect to local deformations for (finite volume hyperbolic) 3-manifold groups, this and this answers your second question positively.
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 here (and references therein).