How can I check via software if my iPhone is unlocked? I have a carrier-unlockable iPhone. That means they sell it locked but will unlocked upon request.

As is generally the case with carriers, mine is a sack of incompetence. I requested the unlock and got many different responses and directionless guidance in the process. I have reason to believe it may be the case that they did attempt to unlock it at some point.

I'd like to know how I can check wether my device is still carrier-locked or not.

I'm on a Mac, and I'm enrolled in the iPhone developer program. It would be of more general use to have an answer that doesn't depend on the latter, tho.

While "just pop in a SIM card from another carrier" is a technically feasible answer, I'd rather have one that doesn't involve my getting off my chair.
So in the end I had to do the SIM card dance.

In fact, it seems what the carrier does is mark your phone as allowed for unlock.

Then, the first time you put a SIM card from another carrier, you must connect to iTunes, which will then perform a series of annoyances and alerts and whatnot, and only then your device will be truly unlocked. You need an internet connection for this process.

I know this is a terrible answer to my own question.