Anyone got any suggestions for handling a character that wants a pet dog? I'm working hard on nixing the Hellhound puppy idea, but I feel like I should let her at least get away with a Rottweiler.

Something like treat its intuition as a pilot rating, and - if the owner is nearby - allow them to make a teamwork test based on "animal handling"?

I've had a falconer in my campaign before that used the falcon (peregrine, IIRC). She had a skill, animal handling: falconry for a specialization (this was SR3 specializations). She would roll that to give it instructions basically exactly as you said. We pre-determined what learned behaviors that the falcon had which were basically to circle around a target (for long-range visual spotting) and to attack (because throwing a bird of prey at someone is cool!). The character was a mage and later researched and designed a specialized translate spell for use with animals as an ordeal. Then she and the falcon could have more in-depth conversations. I don't remember the stats on the spell, but I do remember the the hits on the spell provided extra dice to the animal's chance to understand what their handler wanted from them.