Tree wrap on the bark to protect from urine or use a twelve to twenty four inch diameter, three foot high cylinder of hardware cloth around each one.

What a wonderful project! I'm in Southern California, so I can't make any specific recommendations about tree varieties. However, as a regular visitor to our own local dog park, I know how very important LOTS of shade is to the dogs and their guardians. Our dog park does not have enough shade trees - and it can be hard on the dogs in the summer. So best case scenario, in my opinion would be trees that develop huge sheltering canopies in a relatively short amount of time. Also - could one of the reasons for the trees not making it in the past be related to lack of adequate water during the period of time in which they were getting established? I know our parks department cannot be depended upon to provide for special needs such as these. The sprinklers in our dog park are on a set timer (to water the grass). Any tree that might be added to the park would not get any special irrigation and would have to make it on the shallow waterings which (barely) maintain the grass.