Can I legally adopt my stepchildren (in Texas) if the father is still alive and involved (however loosely that may be)?

No, you are not required to legally adopt a stepchild to add them as a military dependent. You just get a special request chit and write "request command sponsorship for my wife and stepchild". Attach marriage certificate and birth certificate. Have your CO sign it. Then take it to personnel and update your DEERS. Takes about an hour, depending on how long the line is.

However, to answer your question: If the bio-dad has any sort of joint custody, then he would have to agree to any adoption. After the adoption was finalized he would have zero rights to the child. That does not mean that you couldn't let him be involved in their lives as much as you want, it just means that the decision would be totally up to you (and your wife) if/when you want to let him see the kids.

A child can only have one legal father. Not two. Adoption makes you the LEGAL FATHER in all ways. He must relinquish any legal claims.

But you know what.. just because you adopt them legally, does not mean you can't ALLOW him to remain in his life. Win Win.

You would need the biological fathers permission and there's no way around that. Sometimes dads may be inticed by virtue of adoption because it no longer holds them accountable for child support. Often this fact is used a bargaining chip.

"can I adopt with his permission, and him still be involved?"

Again, only with the dads permission.

The real dad has to sign off, yes - even if he doesn't have court ordered custody, etc. If the father is known, he MUST sign off. He relinquishes all rights as a parent in order for you to adopt.

It is possible. However you should remember when you adopt a child it becomes yours the same as if you shared DNA. If the marriage fails you would be supporting this child. The child has an equal right to inherit from you.

If he is willing to sign the papers then you can. He might be willing to do it because if the children are adopted then he will not have to pay child support. If he isn't paying child support now then the mom was stupid.

Only with both living parents permission. But try it. The father might consent as it will relieve him of any responsibility for child support.

The father will have to either voluntarily relinquish his parental rights or be found unfit by the court.