Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes.
It seems to me that there is nothing special about the coordinate axes, so indeed if some direction $v\in \partial B(0,1)$ is fixed, we must have that $u$ is (A.C) on a.e. segment of line with direction $v$. In fact, let $T$ be a orthogonal map which sends $v$ to $e_1=(1,0,...,0)$.
Let $w(x)=u(T^{-1}(x))$. Then, $w\in W^{1,p}(\mathbb{R}^N)$ and $w$ is (A.C) on a.e. segment of line with direction $e_1$. By definition of $w$, we must conclude that $u$ is (A.C) on a.e. segment of line with direction $v$.
My questions are:
1) Is it true that a Sobolev functions is (A.C) for a.e. segment of line? If so, is my argument correct?
2) Can this be generalized for a general family of curves? I mean, assume that $\Gamma$ is a family of (Lipszhitz?) curves. How can I know if $u$ is (A.C) with respect to this family, i.e. $u\circ \gamma$ is (A.C) for $\gamma\in \Gamma$? Is there any kind of measure $\mu$, that we can assign to $\Gamma$, in order to say something like this: $u$ is (A.C) $\mu$ a.e. $\gamma\in \Gamma$?
(1) Yes, your argument is correct. The fact that composition with $T^{-1}$ preserves Sobolev classes also needs to be proved, but the proof is immediate from consideration of what this composition does to Cauchy sequences (wrt $W^{1,p}$ norm) of smooth functions.
(2) Yes, and this generalization is one of fundamental results for the theory of Sobolev functions on metric spaces (which lack the notion of a line segment). The statement is: for every family of curves $\Gamma$ the composition $u\circ \gamma$ is absolutely continuous for all $\gamma\in \Gamma\setminus \Gamma\,'$, where $\Gamma\,'$ is a family of curves with $p$-modulus zero. Obviously, this leads to the question of what is the $p$-modulus of a family of curves... from many available sources, I'll refer you to the survey Sobolev spaces on metric-measure spaces by Hajłasz, specifically Chapter 7 and even more specifically Theorem 7.13.