Do you have balanced headphones? If so, which ones?
No, but I will.
If not, are you looking at balanced headphones or a balanced amp in the future?
For balanced headphone output, how important is it to have dual three-pin XLRs as well as a 4-pin XLR?
Not important at all. I'd use the amp's topology as a basis for a consistent cabling strategy. One of the benefits of not being there yet.
For a balanced amp, how important is it to also have single-ended outputs?
Not important, but I would not pass up the chance to use a SE output if I liked it.
If single-ended is important, is the quality of the single-ended outputs as important as the balanced outputs, or is this more of a “utility” thing so you can use all kinds of different headphones?
Well, I don't want it to suck, and that's basically it. To elaborate: if the particular quality and the sound signature of the amp can be enjoyed from either of two different output methods, even if one is clearly superior or has more to offer, that's the balanced-out-with-single-ended-too that you want to build. Provided, of course, we can't have it all. Because you know we want to.
'Cause I wanna be all like "OMG I tried the balanced out and it's like the smoothest chocolate, then I plugged into the 6.3mm and it's the richest vanilla, so you know what I'm doing now??? Wearing all my headphones and ordering more ears!!"