How cool that you are interested in taking in this cat! And how awful that some people can be SO NEGATIVE on here! It astounds me actually.
It isn't a big deal to take in a stray. Stray cats are some of the best pets! My ex is a vet and his clinic works with several rescue groups that place stray cats and dogs in new homes. I've never had anything but a stray cat. So I can say with experience that it is possible, it can be a great thing and I really wouldn't worry so much about diseases and cost. Think positive!
You should take the cat to a vet and have it all checked out though. And get its shots and spayed or neutered like you had planned.
Yes it's possible for your dog and a cat to get along. Let your cat live in a separate room if possible for awhile, and introduce your dog slowly. I'll bet your dog it just curious and wants to play. The cat might not like that. My dog always wants to play with and jump on a new cat. They eventually get used to each other. Sometimes they still chase and get in little fights. But overall it's no big deal.
If you put the cat in one room at first with the litter box (try to do this where you always will be keeping the box) then it will get used to going in the box. Cats are fairly easy to litter train. Also, my cats never go outside. They don't need to. We have too many raccoons and coyotes and foxes. They'd be killed within the year if I let them out.
I say go for it! You're doing a great thing for the cat and for yourselves! Sounds like fun!
P.S. I have three cats (all were strays) and they all sleep at night with us. And they sleep during the day too! I think cats sleep about 20 hours a day on average. Not positive about that but I think so.
And please don't pay attention to negative nellies.