It depends on the individual dog, and, to a lesser extent, to the individual cat. The group you are working with (I am assuming an adoption group) will at test their dogs, which helps, but isn’t fool proof. Dogs have been known to lie on their cat tests The cat test will eliminate the obvious high prey dogs. I had five cats when I adopted my first grey. She was an older dog (7), very calm, and pretty much ignored the cats. My cats ranged from confident to a spook. My second grey was supposed to be cat safe, and he was with confident cats that were used to dogs, but I had to teach him not to chase the spook.
So you see what I mean, different animals react differently with each other. Lots of people have multiple combinations of cat(s) and grey(s), it can work.
I have no experience with puppies, so I can’t answer that