Above the electroweak scale, is the Higgs ever effectively massless? (thermal field)

So before the Higgs attains a vev (i.e., above TeV scales), does the Higgs doublet become effectively massless?

The Higgs doublet $Phi$ (not the physical higgs about the nonzero vev) gains an effective positive thermal mass in the presence of early plasma in the universe that modifies its dispersion relation, and the $mu$ parameter has an opposite sign above some critical temperature (for sure we cannot neglect the quartic higgs doublet coupling $lambda$ too but by definition it is not a mass term above the electroweak scale), so does that mean that there exists a critical temperature $T_c$ where the Higgs doublet is effectively massless (i.e., do these two contributions cancel each other out)?

On another note, does that also mean that the Higgs travel at $c$ at the point this condition is met?

What about the zero-temperature contribution? is the Higgs physical mass at zero-temperature always 125 GeV, even at high temperatures? Does it mean when we compute feynman diagrams (specifically loop integrals) at high temperature, where the thermal masses go as $T^2$, we can justify neglecting the mass of the Higgs doublet in it’s propagator?

Thank you for clearing up my confusion!