There are lots of members here who have both dogs and cats who peacefully coexist or are pals. I think you have a good plan to let them sniff each other out through the screen door and see how they react.
We do not have a cat in the house, but we have a neighborhood cat who thinks he is a dog. I tried introducing Max, who didn't like that the cat stared directly into his eyes. Cats stare, dogs think staring is rude and aggressive. Max jumped around trying to act menacing and the cat was completely unimpressed.
