  We study the (local) Minkowski content and the (local) fractal Euler
characteristic of limit sets $F\subset\mathbb R$ of conformal graph directed
systems (cGDS) $\Phi$. For the local quantities we prove that the logarithmic
Ces\`aro averages always exist and are constant multiples of the
$\delta$-conformal measure. If $\Phi$ is non-lattice, then also the non-average
local quantities exist and coincide with their respective average versions.
When the conformal contractions of $\Phi$ are analytic, the local versions
exist if and only if $\Phi$ is non-lattice. For the non-local quantities the
above results in particular imply that limit sets of Fuchsian groups of
Schottky type are Minkowski measurable, proving a conjecture of Lapidus from
1993. Further, when the contractions of the cGDS are similarities, we obtain
that the Minkowski content and the fractal Euler characteristic of $F$ exist if
and only if $\Phi$ is non-lattice, generalising earlier results by Falconer,
Gatzouras, Lapidus and van Frankenhuijsen for non-degenerate self-similar
subsets of $\mathbb R$ that satisfy the open set condition.
