Definitely don't try to jumpseat with your dog. Even if you list and show with paperwork, it makes enough extra consdieration for a person like me to say no.
It would stink to show at some cargoport to jumpseat with your pet just to wind up at the bottom of the air-stairs without a ride. I just turned 40 myself, so maybe my latent laziness is turning me into a curmudgeon.