Regarding question 1, I don’t recall any lore evidence of a death knight losing their memories after being raised. Outside of what Sarestha said, that is, about the starter zone. Though that lack of memory might be more on part of them being brainwashed intentionally and controlled by the Lich King.
It’s something I’ve seen RPed often and honestly it’s one of those fanon lore tidbits that just feels really right and at home in the setting. I think perhaps maybe there’s evidence of non-death knights being raised without memories but I’m drawing a blank atm. There’s the Rotbrain from Deathknell, but I think they were just opposing the idea of being on Team Sylvanas. But I still say go for it.
For question 2, if one did lose their memory, I feel like they’d receive a new name based on their talents and accomplishments. Wrathplague for instance might be known for their aura of disgusting plague magic around them, or Icebane might freeze everything to the touch.
Death knights are like WoW’s evil, dark version of Jedi. They’re elite knights who all seem to have their own special spin on things and deep backstories. So that might be a fun way to contemplate a name.
As for question 3, I’m inclined to agree with the above posts. Raising the dead seems to scale in cost with the power of the individual. A death knight has been given extra power from their first resurrection, and was already a powerful hero before then. Sylvanas’ Val’kyr are already very limited in the scope of their necromancy, as they cannot even raise worgen from the dead.
Now if the Lich King was the one trying to raise them, I think the death knight would just spring up ready to go another round with little trouble. It would probably take a very expensive ritual, and I don’t think the death knight powers would go away after that.
If that’s the route you wanted to go, I think you’d need to put a lot of weight onto how his resurrection happens. Perhaps treat it less like he’s being resurrected as a Forsaken and more like he’s been awakened from a dormant state then recruited by the Forsaken. The Forsaken, after all, aren’t any specific type of undead. They’re a coalition of undead races that cooperate for survival.