Electric and magnetic field lines can intersect with each other. As an example, put a capacitor inside a solenoid. But also, electric field lines can and do intersect each other. And even magnetic field lines can and do intersect each other.
For an example, place two equally strong magnets near each other and hold them and rotate them so their north poles point towards each other.
Now, in the place right in the middle between the two north poles, the magnetic field is zero.
And the field line pointing away from one north pole and the field line pointing away from the other north pole intersect right at that place where the magnetic field is zero.
The same exact thing happens if you hold two equal electric charges in place, the electric field halfway between them is zero. And the electric field lines intersect.
That said, fields lines are not roads and they are not cars. Nothing flows along them (despite us calling them lines of flux), so they aren't a road. And they don't move, so they aren't a car.
They are a graphical representation. The direction tells you the direction of the field at that point. And the density tells you the strength.
The reality is that every point in space has an electromagnetic field. And the electric part points in some direction and has a magnitude and the magnetic part points in some direction and has a magnitude.
We could draw a vector at every point. Or we could draw field lines. Those are both just pictures and each can be convenient at times.
does it mean that the place where electric field is zero there are no net electric field lines?
You have two magnetic field lines intersecting (one from each north pole) and with the direction of each one pointing in the opposite direction as each other (so both pointing towards the point of intersection). But you also have more field lines intersecting that point as well, whose directions go away from that point. If you want each field line to represent a fixed number of Webers then you need an equal number that point in as point out. They can also intersect at poles, but there as well, the same number are pointing out as pointing in, really you draw it as one curve that has another curve that touches it with them both going in the same direction where they touch. Like two circles that touch at a point.
For electric fields they can also intersect at places with charges, in which case they might all be pointing in or they might all be pointing out. So where a charge is located they point in different directions (just like for the zero field locations) but they could all point in or all point out (unlike the zero field location, or the magnetic pole).