Cats and dogs: i've had both adult dogs with new kittens and adult cats with new dogs. The biggest trick is to keep the cat from running, inspiring the dog to chase.
I think building confidance in the cat and watching to make sure the dog know what is and is not acceptable (yes, you may sniff and nose the cat, no, you may not pick it up, shake it or chase it) is the main goal and let them figure out the peaking order from there. As to which first, I don't think it matters much either way.
Chooseing a dog, basically what Castle Key said.
Pick out some breeds that seem to be what your looking for and narrow it down. I really think the key to finding a breed that is right is meeting with a breeder and meeting the dogs in question. Even if you choose to adopt, it is still a good idea to have a basic understanding of who's who in terms of breeds, which gives you the ability to have an idea of behavior and health concerns if your looking at a mutt and can see what some of its orgins are.
If you absolutly fall for a specific breed but still want to adopt a dog in need, PLEASE, PLEASE, PLEASE contact the breed specific rescue group. A good breeder will both introduce you to the breed , and can give you info. Many are also involved in some way or another with their affilated breed rescue.
Another really great option, if your heart is not set on a puppy, is a retired show dog. I would add that service dog rejects are good, but they usually have a waiting list. A dog rejected from being a service dog usually has one or two "faults" that make them ineligable for service, but make them excellent and pretrained pets. A retired show dog is well socialized, trained and usually a very good representive of their breed. We have numorous clients with former show dogs and without exception they have wonderful dogs. The only potential downside, if you see it as such, is the breeder may wish to retain breeding rights and wish to breed the dog/bitch at some point. If this is the case, it should be in the contract and they should pick up both the costs and the work involved.
Oh, yeah, nice puppy RKL. The play bow at the pig picture is adorable.