There are many plants that can cause dermatitis and rashes in humans when they come into contact with leaves or other parts of the plants. Do your dogs have a large area where they roam? It's very possible that the dogs could have come into contact with poison ivy, oak, sumac etc. and transferred it to you when you are handling/petting them.
Those photos of your plant do indeed look like the Tree of Heaven (Ailanthus altissima) that KentPfeiffer linked to earlier :
http://www.bing.com/images/sea...
If those trees on your property are Ailanthus altissima, be aware that they contain toxins that are harmful to dogs:
http://lsesc.net/toxins/plants...