  Sharp large deviation results of Bahadur & Ranga Rao-type are provided for
the $q$-norm of random vectors distributed on the $\ell_p^n$-ball
$\mathbb{B}^n_p$ according to the cone probability measure or the uniform
distribution for $1 \le q<p < \infty$, thereby furthering previous large
deviation results by Kabluchko, Prochno and Th\"ale in the same setting. These
results are then applied to deduce sharp asymptotics for intersection volumes
of different $\ell_p^n$-balls in the spirit of Schechtman and
Schmuckenschl\"ager, and for the length of the projection of an $\ell_p^n$-ball
onto a line with uniform random direction. The sharp large deviation results
are proven by providing convenient probabilistic representations of the
$q$-norms, employing local limit theorems to approximate their densities, and
then using geometric results for asymptotic expansions of Laplace integrals of
Adriani & Baldi and Liao & Ramanan to integrate over the densities and derive
concrete probability estimates.
