Starting from the numerical solution of the k.p description of a mismatched ellipsoidal quantum dot in a nanowire, including a spin Zeeman effect with values of the exchange field appropriate to a dilute magnetic semiconductor, we propose and test phenomenological models of the built-in strain and of the heavy hole, light hole and exciton states. We test the validity and the limits of a description restricted to a ($\Gamma_8$) quadruplet of ground states and demonstrate the role of the interactions of the light-hole ground state with light-hole excited states. We show that the built-in axial strain not only defines the character, heavy-hole or light-hole, of the ground state, but also mixes the light-hole state with the split-off band: even for a spin-orbit energy as large as 1~eV, that induces first-order modifications of properties such as the spin value and anisotropy, the oscillator strength, and the electron-hole exchange, for which we extend the description to the light-hole exciton. These results are relevant for a wide range of nanostructures, from mismatched II-VI and III-V quantum dots and nanowires, to III-V nanostructures submitted to an applied stress and to Si nanodevices with even small residual strains.