> the ratio between collector current and emitter current is called α.
Uh, yeah, but I've never found Alpha to be much use in BJT design.
> this combination sets the bias voltage yes?
No. Look at the whole circuit. If the op-amp "+" is at ground, and the op-amp is the least bit happy, the op-amp "-" input also has to be at ground. The op-amp "-" input is set by the voltage divider R3 R2. One end is grounded. The only way for the center (the op-amp "-" input) to stay at ground is for the other end (tied to the output) to be at ground. If it is not, the op-amp will waggle its output until it is happy.
The actual output will be "ground" plus op-amp error (few mV) and input signal. Assuming R2=10K, 2:1 divider, if the op-amp has 1mV of error the output will sit at 2mV. NBD. If the input signal is 1V (and neglecting op-amp error), the output must sit at 2V. That assumes infinite gain in the op-amp; a real-world result might be 1.9999V if the op-amp's gain is "only" 10,000.
> I should be adjusting to set my idle current
The idle current IS the "IDC" current. If IDC is 200mA as shown, then the transistor emitter current is 200mA. (Yes, due to Alpha the collector current is only 198mA for Beta=100, another NBD.) The Base current is about 2mA, but could be 10mA to 0.2mA depending what transistor you use. As long as the op-amp output can supply this current, the exact value does not matter.
So there are no bias adjustments. The output voltage IS zero. The transistor current IS IDC.
The choice of IDC is a matter of opinion and is a no-win situation. This amp is VERY inefficient: 5 Watts power for ~0.2 Watts output, ~4% efficient. I have another amp that can touch ~20%-40% efficiency, and of course any Class AB chip-amp can exceed 50%.
The minimum value of IDC is the maximum peak load current. If you want 3.2V peak in 32Ω, 100mA peak, then IDC must be at least 100mA. That gives over 100mW, which should be generally ample. If you actually get up to 90-99mA peaks there will be some rounding as the transistor current falls very low. Since this thing is a power-pig to begin with, I arbitrarily wrote 200mA, just to be sure of very-clean at 100mW-200mW power levels, and 500mW before THD degrades. But this would be 6 Watts per channel total dissipation, which is hard to manage. My 100mA idle headphone amp uses one heatsink from a 70W speaker amp, and still runs quite warm. Gonna need a lot of aluminum to run 200mA at 30V total supply. 100mA should be very ample and half the heat.
But headphones come in a wide range of impedances. For the same power, 300Ω phones need 1/3rd the current of 32Ω phones. If you only drove 300Ω cans, 30mA-60mA migth be a wiser bias, and a lot less heat.
Also note that for the same power, 32Ω needs 1/3 the Voltage of 300Ω phones. +/-15V is gross overkill for 32Ω headphones. +/-7V is plenty. So for a 32Ω-only amp, lower supply rails mean much less heat for the same performnce.
If we want to drive "any" headphone, we need both high current and high voltage. But we do not need both at the same time. Therefore the "IDC" part does not have to have low voltage drop at high current. In fact it can be a simple resistor. I like that: less to wire, and I trust resistors much more than active current limiters. More reliable, and less likely to have odd little nonlinearities.
Using the suggested +/-12V supplies, a 50Ω resistor in place of IDC gives 12/50= 240mA idle current. To estimate the peak output current, kill the transistor and see where the output sits. 12V * (32Ω/(32Ω+50Ω)) or 4.6V 146mA peak, 330mW RMS max output. This will generally be more than ample. For 300Ω loads, 12V * (300Ω/(300Ω+50Ω)) or 10.2V 34mA peak, 170mW RMS max output, which is still generally ample. Total dissipation is 5.76 Watts per channel. If you use +/-15V supplies, I would use a bigger resistor to keep the 32Ω power about the same yet not let total dissipation rise much. 75Ω or 100Ω are resonable values.
The classic current limiter these days is an LM317 with a resistor up its tail. 6Ω to 20Ω are good values depending how hot you want to run. The LM317 is a very complicated high gain amplifier with all the high frequency nonlinearity that implies; it probably is completely swamped by emitter action but I still like a simple resistor load rather than complicated amplifiers.
> I'm pretty sure there won't be dc leakage as I'm running off the ina134.
The INA134 itself adds a small and negligible DC error. I would not worry.
But the same thought applies: is any DC crap leaking out of your source? Most don't. If you know none of your sources leak DC (or subsonics), then omit all coupling caps. Myself, I sometimes have to plug-up strange gear, and sometimes with huge subsonics that will flutter headphone diaphragms to no good effect, so I like a 17Hz cutoff. (I don't try to judge "balance" on 20Hz-40Hz tones in headphones on location, so the small response droop is not a problem.)