Is it permissible to propose to a girl who is thirteen years old, and is this matter considered religiously, socially, and legally acceptable, and is a girl of this age capable of making the decision to agree to marriage?

There is no objection to marrying a girl with a significant age difference if she is of good character and religious commitment. Evidence for the permissibility of marrying a young girl is found in the Almighty's saying: (And those of your women who have despaired of menstruation, if you doubt, then their period is three months, and also those who have not menstruated...) ( divorse/4), and the marriage of the Prophet, peace and blessings be upon him, to Aisha, may Allah be pleased with her, when she was six years old. If the girl has reached thirteen years of age, her permission is required, based on the Prophet, peace and blessings be upon him, saying: "A non-virgin woman may not be married until she is consulted, and a virgin girl may not be married until her permission is sought." If she has not reached that age, only her father may give her in marriage. It is recommended to seek the permission of a girl who has reached nine years of age as a precaution.