I'm actually using exactly this combo at the moment and I'm really happy with it. Unfortunately I don't have any other headphone amps to compare the Aune T1 to (yet), but it does sound better than the headphone out of my Sony AVR with a bigger soundstage and a bit more punch.
Regarding your questions:
1) Yes, and much more so than comparable amps in this price range. I run mine in high gain, but that should not be taken to mean it's not powerful enough with 0 gain, it's just my subjective preference.
2) Yes, but you will be wasting electricity and slightly reducing tube life (which doesn't matter that much since tubes are rated to last 10000+ hours and they can easily be replaced).
3) Yes. Since the HE-500 are very efficient, they will get plenty loud out of any headphone jack. So it's not just about volume, here a headphone amp is primarily about improving the quality of the sound and getting everything out of your headphones. To an extent it is also about altering aspects of the sound to your liking by switching the tube with a different model ("tube rolling").
Don't forget that the HE-500 is fully open and is almost as loud on the outside as it is on the inside (someone on the HE-500 thread joked about using them as speakers). As such in a nighttime situation you might disturb others even on low volume.