1.- You are considering $v(t)$ as the position vector of each point of the curve. Irrespective of whether or not the parametrization is arc length, we have:
$v(t(s))=\bar v(s)$, so is, for each $s$ there is some valor for $t$ being $v$ the same as $\bar v$ is. e.g.
$t(s=1)=1/\sqrt{2}$
$v(1/\sqrt{2}) = (\cos(1/\sqrt{2}), \sin(1/\sqrt{2}), 1/\sqrt{2})=\bar v(1)$
2.-It is a natural way to describe de curve and, as you say, the speed is always $1$ for arc-length parametrized curves. From here, some formulas describing properties for curves are very simplified, e.g. the Frenet-Serret formulas (compare expressions for each parametrization).
In General Relativity the usual way to describe the world line (the curve a particle follows in the space-time) of a particle is parametrizing this world line by the particle's proper time, that is, the arc length of the world line in the space-time geometry. In this parametrization, the magnitude for its four velocity (the "speed" of the particle through the space-time) is always c, the speed of light in vacuum, or $1$ measuring time in distance units.
