If a face-down creature has a "When this dies [...]" effect (i.e. because of Ixidron), will that effect activate when it dies?
Short answer: Yes, if it had one, but the creature lost that ability when it was turned face-down.
"When this dies" is a leaves-the-battlefield ability [603.6c], and leaves-the-battlefield abilities trigger based on their existence, and the appearance of objects, prior to the event rather than afterward [603.3d].
So it doesn't matter what abilities the creature would have if it was face-up or will have in the graveyard; all that matters is what ability it actually does have. Only leaves-the-battlefield abilities the face-down creature has just prior to dying will trigger when it dies. Normally, that wouldn't be any, but some cards (such as Cauldron Haze) could grant them some.
It will not, if that ability was granted from some rules text on the card. It will if something else is granting it that ability after Ixidron turned it face-down (Cauldron of Souls).from the CR:
707.2a If a face-up permanent is turned face down by a spell or ability, it becomes a 2/2 face-down creature with no text, no name, no subtypes, no expansion symbol, and no mana cost. These values are the copiable values of that object’s characteristics.
There are no turn-based actions that instruct you to turn a creature face up when they go to the graveyard.
110.6. A permanent's status is its physical state. There are four status categories, each of which has two possible values: tapped/untapped, flipped/unflipped, face up/face down, and phased in/phased out. Each permanent always has one of these values for each of these categories.
110.6c A permanent retains its status until a spell, ability, or turn-based action changes it, even if that status is not relevant to it.