The answer is likely going to be provided through a TX criminal statute. In some states a person may be required to pay the costs of a hospital test if they requested it but not if the test was performed at the direction of a law enforcement officer for his own purposes. There are some good DWI lawyers in Austin. Consult with one for more specific guidance under your state's laws.
I am only licensed in the State of West Virginia. Comments offered in my answers are intended strictly for the purpose of sharing ideas that person asking a question may want to consider. Legal advice can only be given by an attorney licensed in your state and offered in the context of the attorney client relationship. No attorney-client relationship is offered or created by my answering this question.
Likely not, based on the facts you give. If you were injured in an accident (with NO suspicion of drugs or alcohol involved), you would still be required to pay for the lab tests they did at the hospital. Although you don't say explicitly, I get the impression that you were unable to give the hospital your personal/insurance information yourself? If that is the case, I would suggest talking to an attorney right away to discuss your specific situation.
Yes, notify the hospital of the mistake and the law enforcement entity.
You should NOT have to pay for law enforcements evidence. There is no indication in the facts that an accident occurred therefore you were wrongfully charged
I hope this helps. It's the best answer I can give with the facts in the question.