  We report $^{35}$Cl NMR, ESR, $\mu$SR and specific heat measurements on the
$S=1/2$ frustrated kagom\'e magnet kapellasite,
$\alpha-$Cu$_3$Zn(OH)$_6$Cl$_2$, where a gapless spin liquid phase is
stabilized by a set of competing exchange interactions. Our measurements
confirm the ferromagnetic character of the nearest-neighbour exchange
interaction $J_1$ and give an energy scale for the competing interactions $|J|
\sim 10$ K. The study of the temperature-dependent ESR lineshift reveals a
moderate symmetric exchange anisotropy term $D$, with $|D/J|\sim 3$%. These
findings validate a posteriori the use of the $J_1 - J_2 - J_d$ Heisenberg
model to describe the magnetic properties of kapellasite [Bernu et al., Phys.
Rev. B 87, 155107 (2013)]. We further confirm that the main deviation from this
model is the severe random depletion of the magnetic kagom\'e lattice by 27%,
due to Cu/Zn site mixing, and specifically address the effect of this disorder
by $^{35}$Cl NMR, performed on an oriented polycrystalline sample.
Surprisingly, while being very sensitive to local structural deformations, our
NMR measurements demonstrate that the system remains homogeneous with a unique
spin susceptibility at high temperature, despite a variety of magnetic
environments. Unconventional spin dynamics is further revealed by NMR and
$\mu$SR in the low-$T$, correlated, spin liquid regime, where a broad
distribution of spin-lattice relaxation times is observed. We ascribe this to
the presence of local low-energy modes.
