I can do my best to answer the first half of your question. Although in mind that field lines are a less formal depiction (simply a visual aid) of electrostatic fields than an actual vector field might be, they still need to follow a few rules.
The first is, as you said, that their density in a specific region corresponds to the strength of the field there. This can actually be connected to flux: if you trace a surface in space, count the field lines passing through it, and multiply by the area of the surface, the result is proportional to the electrostatic flux (taking into account, of course, angles, the accuracy of the image, and other details).
The second property is simple. The field lines are tangent to the electrostatic field vectors at each point; simply put, they point in the same direction as the field itself.
Now, let us use these properties to answer your question. In the region surrounding the $2q$ charge, the field will be twice as strong as and pointing in the opposite direction from (in terms of divergence) the field in the region surrounding the $-q$. Field strength corresponds to line density, so $2q$ has twice as many field lines diverging from it as $-q$ has converging. Since lines can't combine or cross, only half of the lines starting at $2q$ make it to $-q$. The rest do not end.
This should make sense. Remember, field lines can be compared to flux. If we create a closed (Gaussian) surface around our two particle system, we get that the half of the lines not ending at $-q$ leave the closed surface. Tallying up these lines and multiplying by the area of the surface gives us the total flux which, if we recall Gauss' law, must be proportional to the net charge ($q$) of our system. Indeed, this is half of the flux we would calculate if our surface only enclosed $2q$.
If I may be allowed to answer the second half, field lines may represent a few different mathematical concepts. I personally prefer to use them, with an appropriate surface, to calculate relative flux, but they may just as easily be used to represent divergence: whatever the situation necessitates.