Dear Yuhang,
Regarding your 1st question:
(Working in SI units)
RHS:
Integral = JJdt = W/m^2 * W/m^2 * sec
Therefore, V/(k_B*(T^2)) * Integral
= m^3/ [ (J/K) * K^2 ] * ( W/m^2 * W/m^2 * sec)
= W/(m^2 K) = units of thermal conductivity.
Best,
Mario