I know that ahadith says that a girl is mature at 9 (as long as she doesn’t have any mental defects). But I also know that a girl can get married just with her father’s permission, and doesn’t necessarily need him present.

I’m not questioning the Imams AS, but could a 9, 10, or 11 year old get married just with her father’s permission? I guess I can kind of understand a 12+ girl getting married with only her father’s permission, but under that, I’m not sure.

Can she still get married at a young age just with her wali’s permission? I guess (as long as he’s a righteous and modest believer) he knows best and is capable of giving her permission if he believes it’s okay, regardless of her (9+) age.
Is that correct?

According to Ayatullah Sistani ( Islamic laws) as long as marriage age concerns its not an issue that when a boy and girl are going to be married but the if anyone has doubt about the consent of father ( or legal guardian) concern even if the girl and boy are not reached to the Bulugh ( puberty age) at the time of their marriage and later on if they are not agreed on their nikah, then they can nullify it. Therefore there is no such forceful law and exploration or discrimination exists in Islam.
Agha Sistani's Verdict:  A father or paternal grandfather can wed to someone his non-bāligh child/grandchild or his insane child/grandchild who has become bāligh while in the state of insanity. After the child becomes bāligh or the insane individual becomes sane, if the marriage is detrimental for them, he/she can either approve or reject it. But if such a marriage is not detrimental and he/she annuls the marriage after they become bāligh [or after the insane individual becomes sane], the obligatory precaution is that they must either get a divorce or conclude another marriage contract.

Also if the girl and boy any of them they have not reached to the of maturity and physicaly not ready for any sexual relationship then its not permissible to have sex the young wife of husband:
 If a father or a paternal grandfather marries his non-bāligh son/grandson to a girl, then once he becomes bāligh he will have to pay for his wife’s living expenses. In fact, even before he reaches bulūgh, if he is of an age when he is able to derive sexual pleasure and his wife is not so young that her husband cannot derive sexual pleasure from her, then in such a case, her maintenance (nafaqah) is his responsibility. Otherwise, maintenance is not obligatory (wājib) on him.