For the TTL NOR gate, won't the 5 V drop across the resistor always be the same, regardless of $A+B$, making $Q$ always the same?

No-- if the transistors are both off, there will be no current drawn from the supply. Then there's no current going through the resistor at the transistors' collectors, so the output will be at $V_{cc}= 5~V$, i.e., the output will be ``true''. If either or both transistors are on, current through the resistor causes a drop across it, and output is low (``false'').