Is it permissible for a young man to propose to a religious girl he is convinced by, knowing that his mother disapproves of her and asks him to break off the engagement? And is the mother's anger in this case considered one of the results of performing Istikhara?

An engagement is merely a promise of marriage and can be broken. Your breaking off the engagement with the girl is not a sin. If the girl is religious and of good character, a small age difference or a hair problem should not prevent marriage to her. Try to convince your mother, but if she refuses, then honoring her takes precedence, unless you fear temptation (fitna), in which case it is permissible to marry her while striving to appease your mother later. It is not a condition for Istikhara (prayer for guidance) to feel a sense of relief or constriction. If the marriage does not take place, it is obligatory to distance yourself from the girl and cut off communication with her, as she is a non-mahram to you.