I am sorry to hear about your dog. Yes - you can sue the dog owners. You cannot sue the insurance company directly under Texas law. The law in Texas had been, for years, that a dog owner could not recover mental anguish damages for the death of the dog. However, a recent Fort Worth Court of Appeals opinions decided the issue differently. The issue is currently in front of the Texas Supreme Court. The law does allow recovery of vet bills.
I cannot tell from your post whether you were married. If you were, the divorce decree might cover the debt. Consult a family law attorney. If you were not married, you might be able to sue. The Statute of Frauds (no written contract) might be a defense for your ex. You might consider filing in small claims court, because you don't need to know the rules of evidence or procedure to litigate a case there.
Others have responded to your questions about your lawyer. However, insurance companies commonly refuse to provide policy limits to me pre-suit. Once suit is filed in Texas State court, if I serve on the opposing lawyer what is called a "request for disclosure," the opposing lawyer is required to produce copies of all relevant insurance policies.
There is no "standard" fee, because any agreement as to a "standard fee" between lawyers not at the same law firm would likely violate federal law. Lawyers, like many other professionals and businesses, cannot legally "price fix."
So what does this mean to the person needing legal services? The fee is negotiable. Some lawyers will never negotiate from what they typically charge clients, while others will negotiate based on the type of case. For example, if a lawyer believes that he or she...
While you can terminate your attorney's representation at any time in Texas, I would not encourage you to do so absent really compelling circumstances. Even if you terminate representation, you might still be liable to pay the attorney's legal fees and/or costs in the future. I would request an in-person meeting with your lawyer. Send a letter if the law office will not respond to your phone calls.