Legal Question in Family Law in Texas

I don't know what it means to "take someone to court for adultery." Adultery is not a crime in Texas.

He may be threatening to allege adultery in connection with a divorce case.

One proves adultery the same way one proves anything else: evidence. Your statements are evidence (since you're a party, it's not hearsay), so stop talking to him about hanging out with people.

He can fight you for custody of your daughter if he wants to.

The death threats are another matter; that's called family violence, and the courts take this very seriously. You should write down the date and time that he made the threat, and the exact words, if you can remember them.

Adultery is not a crime in Texas. However it is grounds for divorce and it is grounds for seeking a disproportionate distribution of the marital estate. A judge could also consider this in a child custody dispute. A divorce court would consider the best interests of the child. Since you are closer to the child than dad is in Iraq or wherever he might be deployed to, he is going to have trouble taking her daughter from you.

Threatening to kill someone is a crime in Texas. You can report the threat to the local district attorney or to the local JAG.