Need Help With Transistor Logic Circuit Design
"I can see the Q3 transistor is forward biased even when all inputs are zero"

I don't see that.  The Q3 base is floating when all switches are off, because Q1 and Q2 are off.

Actually, that isn't quite true.  Since none of the transistors have any resistance or other circuit at their bases to ensure that they turn off when their inputs are removed, the exact state of each transistor is unknown.  There might be enough leakage current through Q1 and Q2 to tickle Q3.

Also, when switches A and B are closed the resistance from the Q3 base to +12 V is only 10 ohms-ish.  If Q3 is not a power transistor capable of handling a 1 A base current, it will fail quickly.