1) The Eos did very well when I tested it at 38 degrees with lithium cells, settling into steady regulated output for four-and-a-half hours on high and fourteen hours on medium. I find medium is plenty bright for a lot of nighttime travel, and use low around camp. My only caution is that some folks prefer a wider beam to the Eos' narrow output. The PT Quad is a worthy option in that case. Either light takes little pack space and is quite light, especially with Li cells.
The Eos output graphs are available in this test report, in case you've not read it.
2) I'd be curious to hear how such an experiment turned out; I've never tried fashioning a headlamp "cozy" myself.
3) I don't know whether a headlamp would be comfortable under a hat, but the battery pack will stay pretty warm under a jacket hood. Even uncovered it'll still pick up body heat, but how much is probably very temperature and wind-dependent. Smallish two-piece headlamps like the BD Zenix aren't very noticeable when worn, but bigger ones like the Apex and Corona are hard for me to ignore. Some folks don't mind them, though, and the Apex has a lot of fans. It's a real powerhouse, that's for sure.