Ok, here's a more updated schematic. The first stage atenuates anything below 12KHz from input audio, so I'm not sure what is meant by 'swings symmetrically' - this is in fact what I anticipated. The second stage is a non inverted amplification of filtered signal. Because the current was not strong enough, I added a LM386 Audio Amp directly after 2nd stage. Measuring the signal on output of 386 yields a very narrow width inverted pulse, which is exactly what is needed, and that is fed into left primary of transformer. Now with the right primary, through sheer trial and error, I was able to get the desired dc bias(not 100%) with a capacitor connected to 2N collector.
That being said, I thought I could hook a single power supply to non-inverted input of 2nd op amp(basically the feed from stage one). Is this possible, or should I use one of the remaining two op amps to create single power supply?