Does displaying a personal photo on a Messenger profile prevent angels from entering the house, and does this fall under matters of al-Wala' wal-Bara' (loyalty and disavowal)?

Such images are a matter of considerable disagreement among scholars, and issues of considerable disagreement do not warrant condemnation. Rather, they call for mutual advice, presenting evidence, and weighing opinions. If the matter escalates to separation and establishing loyalty and disavowal based on it, then this is clear and reprehensible extremism.