My last K9 partner was extremely well behaved and she was the type of dog I could have taken to WDW. Also, having been there numerous times, I would not have minded coming back after 5 hours or so to empty her out and take a quick break. I also clean up after them. My agency goes so far as to give us pick up bags. I suppose I am in the minority. She was a German Shepherd and shed a lot of hair even though I groomed her constantly. My current partner is a yellow Lab and he sheds worse than she did. Dog hair is a fact of life. We keep several lint rollers around for our guests.
Now, as far as bedbugs and dogs, I can tell you that outside of the explosive detection K-9 unit that Disney has, they also have bedbug dogs. While they aren't in the room for long, most of the rooms in Disney have had a dog in there. Granted, there isn't a lot of time for hair to fall, there will be dog hair and some dander in those rooms. More than likely, you've been in a room that has had a dog in it, even if only for a few minutes.