Let's define a mikado configuration $m$ as a countable collection $\{T_j\}_{j \in \mathbb{N}}$ of disjoint subsets of Euclidean 3-space $(\mathbb{R}^3,\cdot)$. Each $T_j$ is a "tube of radius $R>0$" (independent of $j$). That is, for every $T_j$ there is $x_j \in \mathbb{R}^3$ (the tube center) and $v_j \in \mathbb{R}^3$ with $\|v_j\|=1$ (the tube direction) so that $$T_j=\{y\in \mathbb{R}^3|\, \|y-x_j\|^2<|v_j\cdot(y-x_j)|^2+R^2\}$$
Q1: Does there exist a mikado configuration $m$ where the tube directions $v_j$ are distributed randomly(*) and the tube density is the same everywhere(**)?
Q2: Does there exist a mikado configuration $m$ where the tube density is the same everywhere and the tube directions are locally distributed randomly(***)?
(*) With "the direction vectors being distributed randomly", one could first demand that the empirical measure $\frac{1}{N}\sum_{j=1}^N \delta_{v_j}$ converges weakly to the uniform measure on the unit sphere $S^1 \subset \mathbb{R}^3$ (for the induced topology from the Euclidean topology in $\mathbb{R}^3$)
(**) With "the tube density being the same everywhere", one could intend that the fraction $\frac{\text{Vol}\left((\cup_{j}T_j)\bigcap B(x,r)\right)}{\text{Vol }B(x,r)}$ converges to an $x$-independent constant $c\in (0,1)$ as $r \to \infty$.
(***) This could mean that the empirical measure $\frac{1}{N(x,r)}\sum_{j\in \mathbb{N},\,T_j\cap B(x,r)\neq \emptyset} \delta_{v_j}$ converges weakly to the uniform measure on the unit sphere $S^1$ as $r \to \infty$.
About my own efforts to solve this problem: I thought about using a variant of Olbers paradox to negatively answer those questions. The reasoning is that if we place ourselves in the middle of one of the tubes of one of those supposedly-existing configurations and we look around us, we would see a sky filled with tubes everywhere. If we remove the tube from where we're looking, that situation doesn't change. If we then want to reinsert that tube, we fail because our tube hits other tubes in every angle of the sky. Contradiction