You should not have left w/o a copy of the purchase documents. Now you have to sue, without the documents. A call to the AG's consumer...

It would take an action in partition to resolve the ownership issues.

File a replevin action against the person who has the dog, assuming you have documentation of your ownership.

Send a written complaint by certified mail. If there are further intrusions, retain counsel to deal with him.

Discuss with your AG's consumer protection division.

Send a copy of your vet's report to the shelter and its vet.

Moving to lawsuits and disputes, which is more appropriate. You need a necropsy showing the cause of death was the poison spread by the pest...

Moving to landlord/tenant, where this belongs. Retain counsel if you wish to challenge the eviction.

This site is for American legal issues; not British ones.

If your animal is loose; you are the problem - not your neighbors. Read the citation, which will set forth the options.