Can I sue my ex for damages and cheating on me before we divorced?

Texas is a 'no fault' divorce state.  This means that regardless of the reason, either side can file for divorce.  Texas does allow one spouse to claim adultery as a basis for divorce... but it usually has little effect on the outcome of the divorce action.  Hurt spouses have spent $10-50k plus dollars in divorce actions to have their spounses affairs exposed, only to learn that more and more judges really don't care about the reason-- they just need to know that the marriage is over.
An affair can be relevant to a divorce action if it relates to custody or property.  The courts will consider it relevant to custody if the new paramour is abusive to the child the subject of the divorce suit.   So, if your wife's new boyfriend is beating on your daughter, sexually abusing your daughter, or is yelling at your daughter all the time, then you may be able to take your wife back to court to modify the custody arrangement.  You will need to show there is an emergency situation that represents an immediate harm to your daughter because it has been less than one year since the entry of the divorce decree.
With regard to property, if you can show that she somehow defrauded the community estate by making big gifts to her boyfriend and you had no knowledge of it during the divorce, then you may be able to re-open the divorce for the limited purpose of addressing those issues.
Even though there is potential for re-addressing those two issues, there is little potential for re-opening the divorce just to expose the affair.  The court will simply say that adultery is a basis for granting the divorce, the divorce has been granted, so that issue is now moot.
I know you are hurting because of the conduct of your ex-wife.  However, you need to keep your own emotions in check for the sake of your daughter.  Some spouses begin to bad-mouth the other spouse by telling the child about the affair, making comments like 'your mom abandoned us,' and then intentionally or unintentionally influence the child to reject the other parent.  This is called parential alienation.  If you engage in this conduct, then she could take you back to court, make an alienation claim, and attempt to get full custody of your daughter.
Divorce is a rough process.  Often, it's not fair that the good spouse is stuck holding the bag for support, for bills, and the emotional drama that the children go through.  It's not a perfect process... but often... it's cheaper to deal with the emotional aspect in counseling rather than the legal battles in court.  It sounds like you really care for your daughter.  That's a great starting point any time you start over.  Regardless of what your ex-wife does or does not do, you need to make sure that you are always there for your little girl.  If you focus your daughter, rather than what your ex-wife has done, you will move forward and your daughter will have a greater chance of moving forward with you...which is the biggest win at the end of the day.