For a classical double slit, there is a classical electromagnetic field and so there is an electric and a magnetic field at every point in space.
But the phase that is drawn is more like a tracer. If you had a wave like $A\sin((x-vt)/\lambda)$ then it has an amplitude, a speed, and a wavelength. You could draw the place where it is $+A$ and see how that moves in time (it slides in the $+x$ direction at speed $v$) or you could draw the place where it is $-A$ and see how that moves in time (it slides in the $+x$ direction at speed $v$) or you could pick any other point and track how it moves.
The point of drawing the wavefronts isn't to show where an electric or magnetic field is pointing in any particular direction with any particular magnitude. If we are expecting that at any point the wave changes harmonically in time, such a picture would require the wavefronts to move.
But it didn't matter if we tracked the point where it is $+A$ or where it is $-A$ (how could one of those even be better than the other, it's just a choice of coordinate system to have a $+x$ or a $-x$). We could even track any other point. We can draw one picture and just note that points of constant phase are drawn in black.
It's like a contour plot for phase. At every point in space we have a phase and we draw lines every $2\pi$ units of phase. You could think of them as the peaks, or as the troughs, or as any other phase.
When at the intersection points of two wavefronts the electric field strength doubles (interference of the electric field component of the EM radiation)?
Where the two line cross the waves add constructively, if the individual waves were equal in amplitude the total field is twice as big. Though of course, that point in space is still zero at certain times because the individual waves are zero some times.
Following this point on one of the arcs we get the full cycle of interference including the extinction at some point.
I'm not sure what you mean by following it on one of the arcs. If you move to a different point then that point might also have constructive interference, or destructive interference.
This seems to happen without any consequences until the waves "hit" the screen?
The screen is a location where waves affect charges, so now something else can happen. The screen can (and does) react to the incoming wave.