They should have several forms of immortality, also if someone plays their cards it brings them back to life even after they have been killed. Indeed in episode 134 Judai killed a character named
Zure, Knight of Dark World by winning a duel against him (duels are lethal in the dark world and actually it's not just physical death: it's as if you were hakaied by Beerus because just physical death is no big deal for duel monster spirits). So Zure isn't just a monster that was played and killed during a duel by being destroyed by battle: he's a real character who was also a duelist. Later in episode 136, Judai is dueling
Brron, Mad King of Dark World (a character who duels, not a monster summoned during a duel). During the duel, Brron
summons Zure which
resurrects him. Judai clearly
recognizes that's the same Zure, and as soon as Zure arrives he has
Dreadful aura that shows
his extreme desire fore revenge (he hsouldn't feel that way if that was another Zure). In addition to that, Brron precises once more it's the same zure Judai killed and that he can
have his revenge. Which shows that as long as long as their cards exist, whenever someone calls them, monsters can be brought back. This immortality type 8 (they cannot be fully defeated as long as their cards exist) isn't just physical: even the soul and mind will come back.
Also, spirits can
possess (which btw also counts as
immortality type 6) people to force them to play their cards (that happened in ep 14 of GX with Jinzo) even after they have been killed.
They can manifest themselves physically in the human world and killing their physical form doesn't stop them at all.
They're also
incorporeal and
invisible and can use their hax/powers even when they do not have physical forms (this was showed in more than one occasion).
Even while they're incorporeal they can interact with tangible people and objects to attack/hurt them.
They do not need
food/water and can survive in the vacuum of space
They can live seemingly eternally because the Dark magicial girl (who also appears in GX not just in DM) didn't aged at all since Ancient Egypt 3000 years ago (yeah it's 5k years only in 4kids' version but 3k in the original dub)
They can fly
They have
Supernatural awareness and can detect other spiritual entities and invisible beings. Also they can harm souls and other incorporeal beings.
And there are certainly more that I forgot ^^