Here is an idea. Consider the following parameterization, which is supposed to cover the configuration space in question.

$$\mathcal{C}_7:=\left\{\pmatrix{x_k\\y_x\\z_k},\pmatrix{a_k\\b_k\\c_k}_{1\leq k\leq 7}\in{\mathbb{R}^{3\times 2}}^7\,\middle |\, \text{such that conditions 1.-4. are satisfied} \right\} $$

Conditions:

 1. $x_k^2+y_k^2+z_k^2=1$
 2. $\left\langle\pmatrix{x_k\\y_k\\z_k},\pmatrix{a_k\\b_k\\c_k} \right\rangle=0$
 3. $a_k^2+b_k^2+c_k^2=1$
 4. $d(l_i,l_j)\geq 2$ for $1\leq i<j\leq 7,$ where we define the line
  $$l_k:=\left\{2\pmatrix{x_k\\y_k\\z_k}+\alpha\pmatrix{a_k\\b_k\\c_k}\,\middle|\,\alpha\in\mathbb{R} \right\}$$ and denote with $d(\cdot,\cdot)$ the distance between two lines. 

Note that condition 4. can be rewritten as polynomial inequalities. Hence $\mathcal{C}_7$ is a semi-algebraic set in $\mathbb{R}^{42}$.

 The $(x,z,y)$ are the points, where the unit cylinder is tangent to the unit sphere. The corresponding $(a,b,c)$ gives the direction in the tangent space and the lines $l$ are the cores of the cylinders. (Note that $(-a,-b,-c)$ gives the same cylinder.)

The question "*Is $\mathcal{C}_7$ empty?*" should be decidable. Maybe an algorithmic approach could help from here.  

For the other questions the study of an analogues defined $\mathcal{C}_6$, which we know to be non-empty might be worthwhile. 

**Edit:** I wrote a little program that tries to find points in the described semi-algebraic sets. Here's what it found for $\mathcal{C}_6$ (click [here][1] for an animation).
![6 kissing cylinders][2]


  [1]: http://page.mi.fu-berlin.de/moritz/mo/cyl/kissing.html
  [2]: https://i.sstatic.net/2yK2I.jpg