Is it permissible for a Muslim man to propose to the girl he desires without the mother's consent, given the father's approval and acceptance? And is he Islamically allowed to propose at the age of 21 or 22, considering his desire to marry and focus on his studies?

Gaining the approval of one's parents is among the best acts of drawing near to Allah, for in their approval lies Allah's approval, and in their displeasure lies His displeasure. Therefore, convincing your mother to agree to the marriage is better. If you cannot convince her, then the default is that you are obligated to obey her if her refusal has a legitimate Shar'i justification. However, if her refusal is merely due to a whim, or if you fear temptation for yourself, then you are not obligated to obey her.