Is it permissible for a manager who has the petty cash of the institution to take from it what he needs before his salary is released, and then take from it after his salary is released?

The director who holds discretionary funds is an agent of his employer in disbursing them, not their owner. Therefore, it is not permissible for him to take anything from these funds for himself or for others. Nor is it permissible for him to exceed the customary limits of their disbursement within the work. Otherwise, this would constitute treachery and an unlawful consumption of wealth.