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First, I am very sorry that you have suffered a tragedy in your family. You have my sympathies.
With regard to question: Your facts are scarce, but of course, if you are not amenable to what he wants, you must oppose any Motion he cares to make. I assume he has not made the Motion, but is only threatening? If you live in Texas, he may have to file in your TX county, which will have jurisdiction if you have lived there more than 6 months, which may sway him to not bother.
I am a bit confused on your statement that you "allowed the father visitation." Since you have a custody order, was there not a visitation order there as well? (I say this because generally the father would have the right to visitation with his children.) Also, what is going on with the children's names? Was it his last name at one time and did you change it to something else? Or was it never his last name?
In any event, you need to make sure that if you two can't agree on the changes he wants, and if he files a Motion seeking these changes, that you OPPOSE the motion. You have to make sure to read the papers you are served with and file a written Answer opposing, to better ensure that he does not win by default.
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