The answers at this [related question][1] 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]
 2. [Emily Landes][3]
 3. [Melissa Macasieb][4]
 4. [Vicente Muñoz][5]
 5. [Kate Petersen][6]
 6. [Joan Porti][7]

In particular, the answer to your first question is yes.  See [here][8] for torus knots and $n=3$ and [here][9] for the figure eight knot and $n=3$.

For your second question, I recommend reading about tangent spaces to character varieties [here][10] for generalities.  With respect to local deformations for (finite volume hyperbolic) 3-manifold groups, [this][11] and [this][12] 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][13] (and references therein).


  [1]: https://mathoverflow.net/q/234634/12218
  [2]: https://arxiv.org/search/?searchtype=author&query=Heusener%2C+M
  [3]: http://arxiv.org/abs/1009.3323
  [4]: https://arxiv.org/search/math?searchtype=author&query=Macasieb%2C+M+L
  [5]: http://www.mat.ucm.es/~vmunozve/articles.html
  [6]: http://www.math.fsu.edu/~petersen/publications.html
  [7]: https://arxiv.org/search/?searchtype=author&query=Porti%2C+J
  [8]: http://arxiv.org/abs/1409.4784
  [9]: https://arxiv.org/abs/1505.04451
  [10]: http://arxiv.org/pdf/0902.2589v3.pdf
  [11]: https://arxiv.org/abs/1111.4338
  [12]: http://arxiv.org/pdf/1001.2242v2.pdf
  [13]: http://www.math.lsa.umich.edu/~canary/DCVfinal.pdf