I use extra virgin olive oil because that is what we have prenty of in here. For Ω3 I supply max epa capsules from seven seas, the product that is used as medicine for heart and high cholesterol patients. For Ω6 sunflower oil is a cheap and effective option.
I do not know about flax oil and dogs but I do remember years ago some researchers claims that it presents estrogen like effects. The same goes for all soy products, even olive oil was accused for that but then it was droped as mild effect that presents benefit for lowering cholesterol, protecting heart etc. I repeat, all this is info regarding humans.