My separated spouse has a paramour who spends the night with him when I have the kids. Will documenting this help?

Maybe. Judges are human beings and judges vary in their approach. Some judges believe that you are married until they finalize your divorce and they look askew at live-in arrangements even when the children are not present. Many judges, often in more populace areas, recognize that divorces drag on for a long period of time and that a relationship, as long as it is not aberrant or involving the children, is acceptable. Proving that someone is a party animal only waiting for a final decree of divorce before they cut loose can be important.