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:
- $x_k^2+y_k^2+z_k^2=1$
- $\left\langle\pmatrix{x_k\\y_k\\z_k},\pmatrix{a_k\\b_k\\c_k} \right\rangle=0$
- $a_k^2+b_k^2+c_k^2=1$
- $d(l_i,l_j)\geq 2$ 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 for an animation).
