Mixed NPN and PNP H-Bridge DC Motor Driver
No, your circuit will not work at all, so don't even try it.

Let's just for a second leave Motor-Direction unconnected (open circuit) and inspect what will happen if T5 (Motor-enable) is turned on, the voltage at the top of the H-bridge would want to be at about 11V (T5 saturated).

However, if you examine T3 and T4 they represent a current path of 4 Base-Emitter junctions with a forward voltage of about 2.4V.
Therefore T3, T4 would draw current based solely on the limitation of T5 and your power supply and hold the top of the H-Bridge at about 2.4V. If your power supply can deliver enough current then T5 would be held out of saturation. 

T5 will get very hot and eventually be thermally destroyed .....and if T5 short circuits then you would likely destroy either T3 and T4 next.

The Base-Emitter junctions are simply acting as diodes and it makes almost no difference to the problem if you provide a current drive or not.  

If you want a simple circuit like to work, you'd have to use FETs which are voltage driven.