Yes they are legally married unless a Court says otherwise. Frankly, the party in a position to challenge the legal status of the marriage would be the person who married the person divorced on May 12th. Even then, the challenging person's knowledge of the divorce would be key to any challenge.
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No, this is not a recognized cause of action in the State of Texas.
A mediated settlement agreement ("MSA") is typically "set in stone" as you put it, save and except for the existence of a few limited circumstances which have been referenced. Incorrect or incomplete information from YOUR attorney is not one of those circumstances.
If your name is not on the lease, you will have a hard time gaining legal access to the apartment without his allowing you access or a Court order allowing you access. It appears as though he will not allow you access, so the courthouse is your only option. My advice is to file for divorce immediately. While Dallas County has a "standing order" that must be attached to all petitions for divorce, the terms of which are immediately applicable to you upon filing and applicable to him once...
In short, no she does not have to be present for the court to proceed. While it would be in your spouse's best interest to be present, provided the trial proceeds without her, it will likely help your cause. Without your spouse appearing in person, her attorney will have a difficult time refuting your testimony. It is safe to say most judges will not look kindly upon a party not appearing for their own trial. Odds are, this would be a positive for your case.