I am not an expert on this, but I'll tell you what we tell people at the shelter.
For the dogs first meeting you want them to be on neutral ground. You don't want to bring a new dog into Lily's territory (thats how she sees it). When my brother or I got new dogs we would have all the dogs meet outside. I wouldn't bring the dog into their house and when they got the puppies they were behind a fence in the kitchen so they met my dogs that way....not face-to-face right away. If you get a new dog from a shelter or a foster home you should be able to bring Lily along with you to meet the dog. Then you could see if there would no chance for them to get along. We encourage people (almost demand) to bring in the resident dog for a meet and greet.

Now for the cats. I might suggest to handle that almost as you would introducing a new cat. SLOWLY. If possible keep the cats and the newbie separate so they can learn the smells and maybe see under a door. I happen to have a baby gate I use on the "cat room" door and that has helped with introductions. The dog can see and smell kitty but cannot harm them.

As you know, this could take weeks to have peace in the house. I know it took several weeks before my RB Taz would not smack Zoee on the nose when she got too close. Eventually they all learn how to live with each other in harmony. At least my dogs and cats live in harmony. The two cats on the other hand......not so much.

Also, like I said in Dukes last thread, see if you can get a history on the dog to know how they are with other dogs and cats. I know that's not always easy, but sometimes foster homes can give this info.

GOOD LUCK!! I'm sure you will eventually find the perfect match for your home.