There was 5 wires, do you mean bypass it this way?It's either and inductor or a common-mode choke.
If you can work out how many wires were connected, that will tell you. If there are two ends, it was an inductor. If there were 4, it was a common mode choke.
If it was a choke, that is for noise suppression. If you bypass it by soldering wires across the ends, (near up and down in the photo) it's likely that it will work fine without the choke.
Not really. My guess, based on not really enough information, is that the middle pin is the supply voltage, and is the centre tap. The two transistors operate in turn to ground each end of the centre-tapped winding.From all this 5 pads top left one is connected to GND, is it change anything?