We report ³⁵Cl NMR, ESR, µSR, and specific-heat measurements on the S = ½ frustrated kagome magnet kapellasite α-Cu₃Zn(OH)₆Cl₂, where a gapless spin-liquid phase is stabilized by a set of competing exchange interactions. Our measurements confirm the ferromagnetic character of the nearest-neighbor exchange interaction J₁ and give an energy scale for the competing interactions vertical bar |J| ~ 10 K. The study of the temperature-dependent ESR line shift reveals a moderate symmetric exchange anisotropy term D, with vertical bar |D/J| ~ 3%. These findings validate a posteriori the use of the J₁-J₂-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 kagome lattice by 27%, due to Cu/Zn site mixing, and specifically address the effect of this disorder by ³⁵Cl 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 µSR 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.