Do you have to maintain a single class as the achievement suggests, or will hybrid versions qualify?

Also, in order to qualify for the no death achievement in Epic or Legendary, does that mean no deaths in that setting only, as you could have a death in a lower setting and still earn it?

2 I did all 3 without deaths, so don't know. Better to not check.
1 It qualifies for both classes. Sadly, right now it doesn't work in GOG version. At least for me.

According to the achievement list on Galaxy, someone on GoG has accomplished beating Legendary with Warrior, Defender, and Dream, so it seems to be recognized on GoG.

You say you accomplished all three difficulties without dying but Galaxy achievements show no one has accomplished that in Epic or Legendary yet? Did you play offline?
Actually, I see on your activity list you did accomplish those. Congrats. It's easy to get an accidental death.
You must have been the first to accomplish these since the percentage is still 0.0%.

I see also you have "Eternal Embers." I probably won't buy it. Do you like it? I thought the Atlantis DLC was weak. It had a boring storyline.

Yes, I did. It was easier on Normal and Epic, but harder on legendary. Actually, I did that with Eternal Embers potions. Plus had to ask friend for character with special set - Freya. It gives huge CDR, also give autoheal on low health(while damaging enemies). Build based on wave with electorcute damage. Slow, so have to kite all the time. Also have Defending on the way to this achievement.

Honestly, I don't know why this achievement doesn't apply. Before EE release it should be gotten after killing Surtr. If you didn't know: haardcore achiement is gotten on beating Surtr. So 3 times. And i think it's easy to get it on Dream/Spirit, Dream/Nature or Spirit/Nature. And it is really harder if you play melee.

Actually, Eternal EMbers is harder, really. Too many undeads(no leach). Life drain is OP. And if your character has high CDR, you can be healed each few seconds.

PS Main thing for Hardcore achievement is to play on original speed. On triple speed you will surely get accidental deaths. On original speed you have enough time to react. Did Hardcore in the Grim Dawn too as Warlord, just just kiting while thowing shield.