Moving to Privacy, which is far more appropriate. Cameras are your friend. Use them to document their actions.

Stop posting on FB and retain counsel in TX to review the documents and advise regarding litigation.

Moving to lawsuits and disputes, which is more appropriate. Repost the signs, photograph them, and report further acts of trespass and vandalism...

Report the sister's threats, carry a camera (your phone almost certainly has one), and keep your dog leashed.

This is not a relocation/witness protection service. It is highly unlikely you can change your SSN.

A call to the AG's consumer protection division might prove beneficial to you.

Time to retain counsel to communicate to the pound, using words like "negligence," "malfeasance," and "despoliation of evidence."

A legitimate ESA is protected under the ADA. You may wish to have an attorney make that quite clear to her.

"This girl gave me her dog with all the paperwork..." Then it's your dog. Don't let her "visit" or otherwise have access to it.

Criminal trespass is not an animal law issue - moving to criminal defense so you get better answers. Cameras are your friend.