  We use the coupled cluster method implemented to high orders of approximation
to investigate the frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{3}$
antiferromagnet on the honeycomb lattice with isotropic Heisenberg interactions
of strength $J_{1} > 0$ between nearest-neighbor pairs, $J_{2}>0$ between
next-nearest-neighbor pairs, and $J_{3}>0$ between next-next-neareast-neighbor
pairs of spins. In particular, we study both the ground-state (GS) and
lowest-lying triplet excited-state properties in the case $J_{3}=J_{2} \equiv
\kappa J_{1}$, in the window $0 \leq \kappa \leq 1$ of the frustration
parameter, which includes the (tricritical) point of maximum classical
frustration at $\kappa_{{\rm cl}} = \frac{1}{2}$. We present GS results for the
spin stiffness, $\rho_{s}$, and the zero-field uniform magnetic susceptibility,
$\chi$, which complement our earlier results for the GS energy per spin, $E/N$,
and staggered magnetization, $M$, to yield a complete set of accurate
low-energy parameters for the model. Our results all point towards a phase
diagram containing two quasiclassical antiferromagnetic phases, one with N\'eel
order for $\kappa < \kappa_{c_{1}}$, and the other with collinear striped order
for $\kappa > \kappa_{c_{2}}$. The results for both $\chi$ and the spin gap
$\Delta$ provide compelling evidence for a quantum paramagnetic phase that is
gapped over a considerable portion of the intermediate region $\kappa_{c_{1}} <
\kappa < \kappa_{c_{2}}$, especially close to the two quantum critical points
at $\kappa_{c_{1}}$ and $\kappa_{c_{2}}$. Each of our fully independent sets of
results for the low-energy parameters is consistent with the values
$\kappa_{c_{1}} = 0.45 \pm 0.02$ and $\kappa_{c_{2}} = 0.60 \pm 0.02$, and with
the transition at $\kappa_{c_{1}}$ being of continuous (and probably of the
deconfined) type and that at $\kappa_{c_{2}}$ being of first-order type.
