We systematically investigate the ground-state and the spectral properties of antiferromagnets on a kagomé lattice with several common types of the planar anisotropy: XXZ, single-ion, and out-of-plane Dzyaloshinskii-Moriya. Our main focus is on the role of nonlinear, anharmonic terms, which are responsible for the quantum order-by-disorder effect and for the corresponding selection of the ground-state spin structure in many of these models. The XXZ and the single-ion anisotropy models exhibit a quantum phase transition between the q=0 and the 3×3 states as a function of the anisotropy parameter, offering a rare example of the quantum order-by-disorder fluctuations favoring a ground state which is different from the one selected by thermal fluctuations. The nonlinear terms are also shown to be crucial for a very strong near-resonant decay phenomenon leading to the quasiparticle breakdown in the kagomé-lattice antiferromagnets whose spectra are featuring flat or weakly dispersive modes. The effect is shown to persist even in the limit of large spin values and should be common to other frustrated magnets with flat branches of excitations. Model calculations of the spectrum of the S=5/2 Fe-jarosite with Dzyaloshinskii-Moriya anisotropy provide a convincing and detailed characterization of the proposed scenario.