It is likely that dogs dream in similar ways to humans because the structure of the brain of the dog is fairly similar to that of the human brain. In addition to this, when dogs are asleep they can often be seen 'acting out' day to day activities such as running or eating which strongly suggests that they dream in a similar way to people.
Yes, it is common to dream about your day and the people you interacted with in that day, so I would say that I believe that dogs dream about their owners or humans they interacted with in that day. I would say that since a dog's brain is so advanced, it is very likely that they dream like humans.
Yes, I believe dogs dream like humans because they study humans to such an extent that they cannot help but act like them. Dogs observe everything their owners do in order to make them happy. Just because they cannot speak to us does not mean they do not dream like us.
Though I can't say for certain, I would expect that animals dream about situations and people that are present in their daily lives, much like people do. Stresses and joys appear in human dreams and I would expect the same occurs while animals dream. Challenges and obstacles appear in the dreams of humans and it might be safe to say the same goes for animals.