No.
It doesn't. But if all you look at is the electrostatic force, you won't see how the force propagates. An electrostatic force, by definition, is static: nothing changes. That means nothing propagates; you have basically adopted an approximation where the force a charged object will experience at any point in space is already predetermined, and in this approximation the concept of "speed of propagation" of a force makes no sense.
In order to see propagation, you have to look at electrodynamics: what happens when things change. See below.
No. If the two electrons are a billion light-years apart when they are created, then it will take a billion years for either one to feel any force from the other. This is because you are now including a change: a charged particle is produced where there was none before. (Strictly speaking, of course, charge is conserved; the neutron decay doesn't just produce an electron, it also produces a proton--and a neutrino, but the neutrino is uncharged. But we can assume, for purposes of a thought experiment, that the proton flies off in the opposite direction very quickly, so we can ignore its charge when looking at the electron-electron interaction. Or we could have the electron fly off, since it's a lot lighter, and look at the interaction between the two protons.) Thus, there is a change in the source of the EM field, which means the electrostatic equations do not apply; you have to use the more general equations that cover dynamic situations.
Yes. The electrodynamic equations--Maxwell's Equations--tell you that, whenever there is a change in the source of the field, an electromagnetic wave is created that carries information about the change--the change in the field produced by the change in the source. This wave propagates at the speed of light.
