Do class D headphone amplifier ICs typically contain integrated step-down converters?
Headphone drivers (Class D or otherwise) for portables are limited in swing by the power supply. They do however use a few tricks to achieve greater loudness:

(more expensive ones) full H-bridge drive
(cheaper ones) common-mode bass drive

To answer your voltage-conversion question, Class-D drivers use a kind of PWM-to-analog conversion that basically behaves similarly to a step-down DC-DC. As you might infer, being step-down-like means the voltage swing can't be greater than the supply, and it can of course be much lower depending on the signal level.
That said, for your application, consider the common-bass type with analog drive. Class D at the 1-10mW level will not be that efficient.
As far as the headphone impedance, in the big picture, it's more cost-effective to just specify the right impedance for the application. 32 ohms is pretty standard now, and 8 or even 4 ohms are available. The exact impedance isn't so important, so long as the headphone sensitivity is adequate to get the sound pressure level (SPL) needed for listening. That's not such a difficult problem.
Some more background here: https://www.headphonesty.com/2019/04/headphone-impedance-demystified/
And why are there such things as 120 or even 600 ohm headphones? Answer: vacuum tubes. Tubes have high characteristic impedance, so making the headphone load also high impedance avoids the need for a transformer, which reduces cost and (possibly) improves fidelity.
Related but unrelated, let me talk about cars for a moment. The reason why will become apparent shortly.
How do high-power Class D amplifiers for cars achieve such high power (multiple KW) from just +12V? The amplifier has a built-in boost converter to make the higher voltage, which then is fed to the amplifier driver stages.
This same voltage-boost approach could be used to make a driver that could handle a wider range of headphone impedances, although on a smaller scale.
This driver uses a scaled boost voltage: https://www.maximintegrated.com/en/products/analog/audio/MAX98390.html
Here's another: https://www.ti.com/lit/ds/symlink/tpa2013d1.pdf
I haven't seen the next logical step: merging boost conversion to Class D in one stage. I suppose it's possible - modulate a pair of boost DCDC and AC-couple the output. Product idea? Research project?