Will electromagnetism separate into electricity and magnetism as the universe cools down?

In the case of electroweak force and electromagnetism there is an Higgs mechanism, which makes the $W^{\pm}, Z$ bosons massive, and preserves the photon $\gamma$ massless. But the symmetry relating the electric and magnetic fields is actually a Lorentz symmetry, which is global and remains unbroken. The difference between electric and magnetic fields emerges in non-relativistic limit, but it is not tied with the breaking of symmetry.

No. Separating electric fields from magnetic fields would require the breaking of local Lorentz symmetry, not some gauge symmetry, and there is no reason why that is going to happen.