A dog spent about three hours with us today, her name is Caramel. I was watching her for a neighbor while he had to go to an appointment. Based on my experience, people tend to be too pushy and expect nonhuman animals to make friends and get along much too fast, you really have to let them go at their own pace.
Caramel is very cat friendly, she even seems to understand them well. Pepper is not yet stranger friendly, but shows no difference between cat or dog. So I let Pepper observe from her perches, where she feels safest. Eventually she figured out that Caramel could not get onto the counter or windowsill and just watched from there.
Pepper won't claw anything, she hasn't even used her claws to defend herself yet, so I knew Caramel would be safe. When Caramel did have a chance to get near Pepper, my girl took a defensive posture, which is really just a glare with ears back, at the approaching stranger. Caramel sniffed for just a bit then backed away to explore more of the apartment.
That was it. Caramel is willing to be friends but also seems to know to give Pepper her space until she'd ready. I was in control of the situation, though I was allowing it to play out on it's own I could step in before either got hurt if one of them showed any aggressive behavior. That is the key to introduction, let them decide the speed.
Eventually the two will become friends or indifferent. Indifference must be acceptable, you cannot force friendships, but indifference can later become a friendship if they are not pushed together. Younger nonhuman animals tend to become friends much faster, and if they have only positive encounters with strangers the younger ones will be very friendly.
Older nonhuman animals tend to take a lot more time and patience if they have either had a lot of bad experiences, or no experiences to learn from. Pepper is the latter, she doesn't fear nonhuman animals at all, but has no idea what to think about them because her only exposure to strangers was at the shelter where it was a gambit if behaviors.
There is a chance an adult cat will always be a loner, they won't like any strange nonhuman animals, ever. These are also the ones quick to growl or scratch your hand if you ever pet another animal. So it is advisable to pet another nonhuman animal first, get their smell on your hand, then allow your friend to smell your hand.
How they react to this will tell you if they are capable of making friends. For Pepper it was curiosity and indifference, so I knew what to expect their first long encounter. This was why I did not mind watching Caramel at all, Pepper knew Caramel existed before meeting her and I knew Pepper's reaction as well.
All in all, it was a great experience for all of us, Caramel finally got to hang out with me, she has always been excited to see me in the lobby. Pepper got to watch and see what Caramel was about. I got to see enough Pepper's reactions to know her pace for future introductions.
In a few months I may be watching Caramel for a few days, it is possible they become best friends, based on their reactions to each other. Caramel has been around cats all her life, very positive experiences, which is why she understands them so well.