special relativity - Does contracted spring weigh more than stretched one?

(One of examples that potential energy contributes to mass.)

Does hot object weigh more than cold one?

(One of examples that kinetic energy contributes to mass.)

If these are true and justified by mass-energy equivalence principle, could the underlying principle be justified?

I've learned and derived $E=mc^2$ but only for the case that $m$ is interpreted as rest one. Could I generalize this argument to include kinetic and potential energy? :

$$mc^2=E =∑m_0c^2+∑K+V=∑\gamma m_0c^2+V$$