Legal Question in Family Law in Texas

You asked two different questions:

1. Can the father file contempt? The answer is "yes" but he should expect that the mother will file contempt on his child support arrearage. Since it is far easier to prove failure to pay child support than possession, the father is far likelier to be held in contempt. If the father did not show up on time to pick the children up, then he could not win anyway.

2. Can the father get custody? Unlikely if these are all of the facts. It is fairly well accepted that child support does not usually come close to paying even half of the cost of raising a child in most cases. Why would a judge give custody to a father who is not even paying his child support when he would have to pay more if he had the children.

As in all cases, there are almost assuredly other factors that will enter into this that would make these answers potentially incorrect. You need to tell the entire story to an experienced family law attorney for a better estimate of the father's chances.