Is it permissible for the active partner to take work expenses (such as transportation costs) from the company's profits without the knowledge of his partner, and can he claim them for past years?

It is permissible for you to take the cost of your transportation if the work serves the company's interest, and there was no prior agreement for you to bear the expense yourself. However, if the work does not serve the company's interest, or was not agreed upon, then you are not entitled to claim what you spent except with the partner's permission, and likewise if you donated the expense.