No. You don't have to worry as plant mites are species specific and won't suddenly decide that your dogs are tasty.

You could dip the leaves in soapy water to get rid of them on the ivy.

Dump the English ivy. Whatever you do, do not plant it in your garden. It quickly becomes a weed. It totally choked and killed 2 trees in my front yard. It is hard to pull up, once it becomes established.