No, it didn't. Mass is not required for gravity; only energy is. The energy was there before the electroweak phase transition; it just wasn't in the form of rest mass. It still produced gravity.
So you're right technically, but it has nothing to do with what I said in my first comment - without the higgs field the universe as we know it today would be unrecognizable, and a universe without a higgs field would not look like ours.
No. The electroweak phase transition had no effect on gravity whatever; the stress-energy that was governing the expansion was the same before and after. Again, the source of gravity in GR is the stress-energy tensor, not rest mass.
E = MC^2
Can't have energy without mass, and mass leads to gravity.
https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relati...
E^2 = (MC)^2 + (PC^2)^2
The first term is describing the rest mass. You can redefine the mass term to make it E=mc^2 but now this mass does not correspond to the rest mass. And for sure you can have energy without having a rest mass. Actually for the early universe all you had was a form of radiation and energy.
Is only an approximation for particular cases, not a general law.