The details of the electric current is a bit complicated. A simple explanation is that electrons are always whizzing about in all directions and current flow is the net flow of electrons in a direction. The frequency of this net flow is the rate of the oscillation in it. The electrons themselves aren't vibrating back and forth at this frequency.
Well, I'd say that in the context of current flow, the signal is the measurement of the voltage or current flow at any particular moment in time, regardless of its properties. The behavior of the signal can be described as wave-like when it behaves a certain way, namely that there is a repeating pattern that a wave equation can be applied to.
That's right. Mathematically, any pulse can be broken down into the waves composing it by using a Fourier Transform.