How come the provincial + federal tax for my T4 alone is greater than Box 22? I thought box 22 should already pay for all that?

Your tax owing is calculated from your income, any deductions as applicable, and allowing for non-refundable tax credits.  This is then compared to your refundable credits which include the amounts held back from your pay cheques through the year (box 22 of your T4).  If the refundable credits are greater than your tax owing you get a refund.  If the refundable credits are less, you owe the difference  (have a balance due).

How is box 22 calculated? I was under the impression that it is directly the total taxable amount of the base salary (+ bonuses) with the federal and provincial tax rate applied?

Box 22 is merely the sum of monies deducted through the year for taxes.  If your pay fluctuates through the year it is possible that the amount deducted may not be enough.  As to what was deducted from your pay...you would need to ask your employer.