No reason not to bring your dog, many people do. We like you could not bear to leave ours at home or with someone else, so they have travelled with us always. With regards to National Parks most do not allow dogs on dirt trails, but do allow them on any paved trails and at the various lookouts the park may have. I seem to recall a couple of parks that do allow dogs on dirt trails, but they if so are the exception rather than the rule. One ranger told me that dogs are allowed on all trails if carried by the owner in a dog carrier, bag etc. With two large Akita's that does not work for us, but may for you, but as always check with any park, National or otherwise to get their rules etc. As far as leaving in the trailer, you want to make sure your dog will not nuisance bark and you will want to make sure AC is available if hot.