Does conductance in the transmission line model represent a physical quantity?
non-real characteristic impedance transmission lines exist? Are they typical?


Yes. 

An ideal transmission line has R = 0 and G = 0. This gives a real characteristic impedance. But this is an idealization. These numbers are never actually 0 in reality, so the imaginary part of characteristic impedance is never truly 0.


  What does it even mean to have a non-real characteristic impedance?


It means the transmission line is lossy. The sum of output power and reflected power will be (at least slightly) less than the input power.


  but if hope to achieve a real Z0 this means carefully matching the non-ideal properties of both the conductor and the dielectric, 


This isn't practical. 

More often you just try to get R and G low enough that they can be ignored for practical purposes. 

Minimizing G means choosing a dielectric material appropriate for the frequencies you're using.

Minimizing R means using a larger conductor (wider trace on a PCB or larger diameter coax). But this also tends to decrease the maximum frequency before the transmission line goes multi-mode. End result: at high frequencies, you just have to use small geometries and deal with the loss.

Due to skin effect, R tends to increase in proportion to \$\sqrt{f}\$, and this will typically dominate the frequency-dependent loss of a transmission line, if the dielectric has been chosen appropriately for the operating frequency band.


  R and G don't directly correlate to physical quantities,


R is pretty directly tied to the conductivity of the conductor material. Skin effect also plays an important role. This is why you see silver-plated conductors on coax meant for high frequencies. 

G relates to the polarization behavior of the dielectric, so it's harder to tie down to a specific physical mechanism. Generally if you're choosing dielectrics for high frequency designs, you'll see the loss specified as a loss tangent, but it will probably only be specified at a single frequency, which naturally won't be the one you're designing for.