  The Thermal Quasiparticle Random-Phase Approximation is combined with the
Skyrme energy density functional method (Skyrme-TQRPA) to study the response of
a hot nucleus to an external perturbation. For the sample nuclei, $^{56}$Fe and
$^{82}$Ge, the Skyrme-TQRPA is applied to analyze thermal effects on the
strength function of charge-neutral Gamow-Teller transitions which dominate
neutrino-nucleus reactions at $E_\nu \lesssim 20$~MeV. For the relevant
supernova temperatures we calculate the cross sections for inelastic neutrino
scattering. We also apply the method to examine the rate of
neutrino-antineutrino pair emission by hot nuclei. The cross sections and rates
are compared with those obtained earlier from the TQRPA calculations based on
the phenomenological Quasiparticle-Phonon Model Hamiltonian. For inelastic
neutrino scattering on $^{56}$Fe we also compare the Skyrme-TQRPA results to
those obtained earlier from a hybrid approach that combines shell-model and RPA
calculations.
