I’ve had 20 or so cats and 4 dogs, not all at the same time, in my house.
Whenever we’ve brought in a new one, we open the door, let the critter in, and they all sort it out.
We’ve had a few spats that we’ve broken up, but for the most part, everyone gets along.
There’s a pecking order they all know, and we do not. As soon as they figure it out, there’s harmony.
The cat stays in "her" private room for most of the foreseeable future. She has a litter box and feeding/watering dishes in there, plus toys and bedding. She should be perfectly comfortable.
We are going to get a large pet carrier (we need one anyway), hard sides not cloth.When all humans are upstairs together, we'll put the cat in the carrier, and let the animals investigate each other through the slats. We will do that for longer periods of time till the dogs get used to her presence. This may take some time-days, perhaps weeks.
When the dogs no longer seem terribly interested in her, we'll let her out of the carrier. The dogs will be on leashes so if they chase her, they can be stopped.
It may be that she will scratch them on the nose to convey the message, "Back off!" Dogs aren't dumb-they should get the message very quickly. At any rate, it will be a loooooong time (if ever) before all three animals are alone unsupervised.
The dogs from Day One have had a 'dog room' in the basement : A big living space with linoleum floors, their feeding dishes, blankets, toys , puppy pads, etc. They sleep there at night and we put them in lock down there while we're gone. We can use this to give the cat more of an opportunity than she's had to explore the house while the dogs are in lock down. This will also get the dogs used to her as her scent will become familiar through all the areas she and they roam(I hope...).
This may all be for naught as she may yet be claimed by her people, or one of DN's cat owning co workers may get her. Plus DN is looking for his own place. But if not-if the cat is here for awhile-I think given what I've read on this threads plus per what I've googled, this has a pretty decent chance of success.
Wish us all luck! (Man, this is *not* how I expected this AM to start : They found the cat while I was asleep...It was a major surprise to find a cat in the house! DN is 23 and he grew up with cats, DBF had at least one cat while he was growing up. Me, I have never had cats because my mother hates them. :-( Hopefully having 2 people familiar with cats will help.) Again , thanks to everyone who has offered advice.