1). "Do you have balanced headphones? If so, which ones?"
-Yup! At the moment, I've got several sets of balanced cables for my LCD-2s, all of which are being run into SE adapters so they'll work with my Lyr. I plan on getting a balanced cable for my HD650s at some point, and generally plan on trying to get balanced headphones whenever possible in the future.
2). "If not, are you looking at balanced headphones or a balanced amp in the future?"
-See #1. Yeppers!
3). "For balanced headphone output, how important is it to have dual three-pin XLRs as well as a 4-pin XLR?"
-For me, not at all. I'm an utter noob, but when I dug into the differences, the consensus seemed to be that 4-pin was the way to go. So I've consciously avoided the dual 3-pin versions and only purchased 4-pin cables.
4). "For a balanced amp, how important is it to also have single-ended outputs?"
-My initially answer when I read your post was that it's essential. Upon further reflection, I don't think it's necessarily crucial, but it's not far off. I personally have two different XLR to SE adapters, but most people probably don't, and not having the ability to plug the most common connection type into it seems like it would really limit the potential success of the device. I realize that having an SE plug on an amp whose primary claim to fame is balanced operation seems like slapping another pair of teats on a boar, but not putting them on there is going to unnecessarily reduce the utility and appeal of the device. This is, of course, assuming the SQ on the SE port is up to par. But knowing you guys, what isn't excellent?
5). "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?"
-Everybody is going to say quality, and I'm going to agree, because the question you asked pitted raw utility versus sound quality, and by answering "quality," you get to essentially pick both answers. If the question is more along the lines of trying to determine how close the SQ needs to be on the SE vs XLR ports, I'd say it needs to be darn close, if not spot on (minus the natural differences between SE and balanced anyway, of course).
A couple of (utterly unsolicited) thoughts:
-Definitely include a tubed option, either on this or a subsequent amp. I use the Lyr every single day and LOVE it, and a big part of that was my ability to play around with the sound by rolling the tubes in it. I don't know of another option you can include on an amp that allows for a $20 upgrade to produce such a dramatic improvement in the sound, and that's a nice thing to have!
-I know you guys aren't exactly riding the world's largest profit margins on your way to Scrooge McDuck-like cash hoards, but if it would be possible to have some kind of trade-in program for existing Schiit owners, it would be greatly appreciated! :)
-I already have more DACs than I can shake a stick at, and have absolutely zero need for another. However, logic and reason have zero bearing on my whorish gear lust, and the thought of the Schiit DAC has popped up unbeckoned in my mind more than once. So some kind of package deal on the Bifrost (or following DACs) with this bad boy would be a fantastic excuse for me to start another fight with my girlfriend over blowing money on audio stuff I don't really need. (Btw, although that sounds weird, to be clear: I am in favor of a bundle deal. Audio's before hoes!!*)
-Don't know if it would be possible to have XLR pre-outs on the back (kind of like the Lyr has RCA outs now), but that would be especially awesome. Just sayin'.
-That kitchen sink you mentioned did sound pretty awesome...
*Okay, so I said that to be cute. My girlfriend is an absolutely wonderful, understanding creature whom I love dearly. Plus, she'd probably stab me in the gibbledy-bits if she thought I'd ever refer to her in the above fashion.