Is it permissible to inform a girl of one's desire to propose to her in the future so that she awaits you, and is it a condition to inform her through a mahram?

There is no harm in informing a girl of the desire to propose to her in the future. It is preferable for this to be done through a woman. Otherwise, there is no harm in informing her directly, while observing the Islamic guidelines for speaking with a non-mahram woman. If marriage to her is possible during the study period, it is a good thing because it protects one from immorality and allows for early procreation, and studying does not prevent marriage. However, if marrying her soon is not possible, the advice is to leave the matter until its time comes to avoid complications.