Are messages sent to the wife considered a divorce, and is an oath sworn in one of them binding, and is the wife's staying in her family's home commendable?

We advise you to fear Allah and beware of the tongue, and not to swear by other than Allah, for the Prophet (peace and blessings be upon him) said: "Whoever swears, let him swear by Allah or remain silent," and he also said: "Whoever swears by other than Allah has committed an act of disbelief or polytheism." Your statement to your wife, "By my father's head, I will relieve you," does not constitute a divorce because it is in the future tense and a promise. As for your statement, "You will not enter it again," it does not constitute a divorce unless you intended divorce by it. The preference for your wife remaining in her father's house depends on the benefit and what you both agree upon.