You choose the mode as the triggered ability goes on the stack. You can choose a mode that requires targets only if there are legal targets available.

It doesn't matter who has chosen any particular mode. For example, say you control Gandalf the Grey and have chosen the first two modes. If an opponent gains control of Gandalf the Grey, that player can choose only the third or fourth mode.

If the ability doesn't resolve (either for having its target become illegal or because a spell or ability counters it), the mode chosen for that instance of the ability still counts as being chosen.

The phrase "that hasn't been chosen" refers only to that specific Gandalf the Grey. If Gandalf the Grey leaves the battlefield (possibly because of the fourth mode of its ability) and then returns to the battlefield later, it will be a new object with no memory of the modes chosen when it was last on the battlefield.