To answer your first question, I'm a bit confused because you seem to be okay with the existence of$\rho$ which is a geodesic asymptotic to $\xi$. Actually, $[a,z]$ is nothing else than a geodesic asymptotic to $z$, if $z$ is in the boundary.

Maybe you need a bit of clarification though. I guess it depends on your definition of the Gromov boundary. I assume that you use the definion with equivalence classes of sequences converging in the sense of Gromov. Then, for any proper geodesic Gromov hyperbolic space $X$, for any point $o\in X$ and any point $\gamma\in \partial X$, there exists a geodesic ray $\alpha:\mathbb{R}_{\geq 0}\to X$ such that $\alpha(0)=o$ and $\alpha(t)$ converges to $\gamma$ in the sense of Gromov. The proof is basically a use of Arzelà-Ascoli lemma. You can actually define the Gromov boundary as the equivalence classes of geodesic rays, declaring that two geodesic are equivalent if they stay within a bounded distance of each other. This is proved in Section 11.11 of the book you refer too. You can also find a detailed proof of all this (and in particular of the fact that these two definitions are equivalent) in Section 7 of de la Harpe and Ghys's book Sur les Groupes Hyperboliques d’après Mikhael Gromov. You can also look at Boundaries of hyperbolic groups by Ilya Kapovic and Nadia Benakli. In particular, they prove there that without the properness assumption, this statement is still true replacing geodesic by (1,10$\delta$)-quasi-geodesic.

For your second question, maybe you have some specific thoughts, but I can only suggest a reformulation. Letting $\gamma\in \partial X$, $\gamma\in U_{\rho(t),k}(\xi)$ if *any* geodesic from $a$ to $\gamma$ passes within $k$ of the point at distance $t$ from $a$ on the fixed geodesic $\rho$ from $a$ to $\xi$. In other words, the geodesics from $a$ to $\gamma$ and from $a$ to $\xi$ start diverging more than $k$ only after time $t$.
Notice that two geodesics asymptotic to $\gamma$ stay within a uniform distance of each other, so that up to changing $k$, you could also ask in the definition that *some* geodesic asymptotic to $\gamma$ passes in $B_k(\rho(t))$. This answer the question in your comment. Note also that because of this property, the choice of the geodesic $\rho$ is not important.

Maybe the following will help and will explain the terminology shadow. If you think of geodesics as light rays, then put an opaque ball around $\rho(t)$ and a light source at $a$. Then, the shadow will exactly be the part which is not enlightened.

Also, to understand why this is a system of neighborhood, note the following. Take a tree (which is 0-hyperbolic) and take $k=0$. Then, $U_{\rho(t),0}(\xi)$ is exactly all the points in the boundary that will be "after $\rho(t)$", in the sense that a geodesic from $a$ to $\gamma$ has to pass through $\rho(t)$. If you make $t$ go to infinity, then you only have $\xi$ left.