Strange behavior on transformer driver
Without the transformer connected, the only load is your scope probe (probably around 10M&ohm; and 20pF). When you measure between one output and ground this tiny load is enough to pull the voltage down when the transistor is turned off. 

But when you measure between one output transistor and the other there is nothing to pull the voltage down. In fact the scope probe is (weakly) pulling the Emitter of the 'off' transistor up until it reaches the transistor's Base-Emitter reverse breakdown voltage at about 8V. This is why you see short spikes (due to capacitive coupling) and a lower amplitude square wave (due to Base-Emitter reverse breakdown).

To get the desired waveform you need to provide load resistances low enough to swamp out transistor parasitics and scope probe impedance. A 1k resistor from each Emitter to ground should be enough. 

However, even with suitable load resistors you still won't get the same waveform as when the transformer is connected. The transformer reflects voltage on one side of its primary to the other, but with opposite polarity. So when one transistor is turned on and switching +12V onto one side of the transformer primary, the other side will put (close to) negative 12V onto the other transistor's Emitter. This will cause the 'off' transistor to turn on when it shouldn't. 

To work with a transformer you need to either change each output transistor to a PNP type and connect their Emitters to +12V, or connect the transformer center tap to +12V and connect the (NPN) Emitters to ground. In either case the Collectors must go to the transformer. Then the Collector-Emitter voltage won't reverse when each transistor is turned off, but go to double the supply voltage instead.