I agree with PP that environmental allergies are not common at this age. My 1 year old has the same symptoms (without the bloody nose), and he has been around dogs (ours, my parents', my brother's) all his life. We were at the pediatrician last night (he developed an ear infection), and I could hear at least two other babies there with coughs. I would say it's a cold.
Both of my dogs came from rescue organizations, and per the adoption contract, I must return the dogs to that rescue organization if I ever find myself in a position where I can't keep the dog. If, down the line, your child does develop allergies to dogs, I'm almost positive the rescue organization will take the dog back. What's the alternative, right??