A"Triggered" keyword like Ambush (on attachments), inherent abililities like the Rusted Sword and Hooked Trident, and House of Talons Daario _do_ allow you to put attachments on characters that are immune to triggered effects (ITE Littlefinger/Crow's Eye/Arriane) because they are acting on themselves rather than the character they are attaching too?
Well, be careful with this. On the one hand, it is true that if a triggered effect is putting an attachment into play, that attachment can end up on an "immune to triggered effects" character because the "put into play" effect is working directly on the attachment - and only indirectly on character it ends up being attached to. It's the same reason why a "discard an attachment" triggered effect is able to knock an attachment off an immune character.
But be very careful about the wording of the "put into play" effect. Some of them specify that you choose a character, then put the attachment on it. If that is the case, the "put into play" effect will not be able to get the attachment on the immune character because the stipulation in immunity against choosing the immune card as a target is still enforced.
And you are correct on the saving thing. They ignore the effects of the save, so end up being killed. They can only save themselves with their own abilities (like dupes, or if Maester Aemon gained immunity to triggered effects, for example).