Yes, dogs can get Giardia: I had an Alaskan Malamute that had a serious case. The most difficult thing about it was getting a diagnosis. From what I was told, most vets do not see very many cases of Giardia in dogs and so are prone to seek other causes first.
As far as poison oak or poison ivy is concerned, dogs can transfer the urishinol oil from their fur to human skin. I don't think you can keep them out of it even on-leash.
I am not (yet) sensitive to urishinol oil, my brother-in-law is highly sensitive to it. A few years back, my dog got into poison ivy and transferred the oil to him and presumably me. He got a major case on his legs, hands, wrists, ears, nose and private parts. I was spared. We used a paint roller to apply the calamine lotion to his affected areas.
He complained constantly that I was not suffering with him. He was a Clinton supporter so I just kept telling him that "I felt his pain".