Addressing your three inter-related questions directly:
In a classical system, temperature, an intensive quantity, does not measure directly the average kinetic energy (translational or not), which is an extensive quantity. The correct statement is that each kinetic energy term of the Hamiltonian contributes by $\frac12 k_B T$ to the average energy of the system (equipartition theorem). Therefore, the contribution to internal energy per molecule of all the translational degrees of freedom is always $\frac{3}{2} k_B T$. This result is independent of the kind of molecules (mono-, di-, tri-,... atomic) and on the specific thermodynamic phase (of course, provided temperature and density are such that quantum effects on translational degrees of freedom can be ignored.
Things are usually more complicated for the rotational and vibrational energy of the molecules. Even at room temperature, the equipartition theorem cannot be used for some of the corresponding degrees of freedom, and for those degrees of freedom proportionality between contribution to the internal energy per molecule and temperature may not be valid.
In conclusion, in the classical regime, solids, liquids, and gases at the same temperature will always have the same average kinetic energy per molecule. The result is even stronger: not only the average kinetic energy per molecule but also the velocity distribution function (the Maxwellian distribution) of the molecules is precisely the same at the same temperature.