Is it permissible to apply for promotion to a position for which I have expressed my desire, knowing that a colleague has already applied for the same position, and does this fall under the prohibition: "None of you should propose to a woman to whom his brother has already proposed"?

There is no objection to a Muslim applying for a public position or job if he is qualified for it, even if others apply, for competition is permissible if it is free of cheating and bribery; because the Prophet, peace and blessings be upon him, said: "Be keen on what benefits you, seek help from Allah, and do not lose heart." This differs from the prohibition of a Muslim proposing to one already engaged, as what is forbidden in engagement is proposing to one who has accepted another's proposal, unless the other is a transgressor.