For iPhone with the Lightning port, is it possible to connect a USB headset to it via some Lightning-to-USB adapter?
Yes. I have done this myself using the Apple Lightning to USB 3 Camera Adapter.
I was able to use a USB audio adapter to plug in headphones. I had no real need to do this but to prove it works. I have other, more convenient, means to connect headphones to my iPhone.
It's called a "camera adapter" but it will work with a number of USB device types. I tested it with mice, keyboards, and audio devices, they all work. If the sound comes out the wrong audio device then selecting it manually will fix it. Sometimes the iPhone will not switch to the headphones that were plugged in.