As others have mentioned, Sir Zeliek is extremely exceptional, as Zeliek himself demonstrates. Zeliek managing to keep his mind fully intact in his condition is pretty much unprecedented, with most others reverting to their old selves (or something similar to it) only after someone or something else intervenes. As a result, Zeliek is massively outside of the norm.
Bolvar is the only one I can readily think think of who has shown similar qualities (unrelated to actually being able to use the light), when he resisted the Lich King’s personal attempts to break him.
If you are forsaken, it may hurt a lot more since the plague which made them is based on something the demons came up with. If you come back Directly, like DKs are, it should not hurt. That being said, the rules of all DKs apply to an undead who uses the light, they still have to inflict pain in order to keep from decaying. They only need to believe their cause is righteous, which is why Sally Whitemaine is a better example instead of Sir Zeliek because he lacks free will whereas she is fully in command of her senses.
Aside from Zeliek, Draenei death knight player characters do not lose their Gift of the Naaru racial and are, indeed, healed by it and capable of healing other characters with it. If they couldn’t use it, it probably would have been changed to a mechanically identical but differently-flavored racial to reflect the lore.
So, it may matter how they are returned to live than just being undead. The Light does not have any type of feeling towards or against the Undead. The light seems to understand that there are differences between the Forsaken and Death Knights.
There is also the fact that the souls of the undead (Forsaken, death knights, or ghouls) are imperfectly attached to their bodies; the dark magic that sustains them is a buffer that prevents their souls from properly joining with their bodies. This is why undead feel only faint sensations of pain or discomfort from most physical stimuli, and why the Light is so painful to their existence.
Death Knights could be so strong willed or use to pain that they fight through the pain the light inflicts on them. So, it might be possible for more undead paladins to exist as long as they could withstand the incredible amount of pain they would have to take.
Then lastly as shown by Saa’ra a corpse can be infused with the light and be resurrect, but still remain undead. This opens up the possibility of undead Paladins that are lightforged.