I introduced a cat to two dogs but I expect introducing a dog would work the same way. I supervised them the whole time they were together and let the cat get up away from the dogs if she got scared. It takes a while for them to get used to each other but now the cat sleeps on the dog bed with my girl dog and they play.
As far as the cat box goes... I haven't had much luck keeping the dogs out except by putting it up. My cat has her litter box, food, and water on top the dryer and it works out nicely. She can get up, dogs can't.
Good luck! And congratulations on your new addition. 
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Why cycling? Anything [sport] that had to do with a ball, I wasn't very good at.
-Lance Armstrong
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