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.

In some very unusual situations, you may not be able to choose a mode, either because all modes have previously been chosen or the only remaining modes require targets and there are no legal targets available. In this case, the ability is simply removed from the stack with no effect.

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

Yes, if the fourth mode is the only one remaining, you must choose it. You read the whole contract, right?

The phrase "that hasn't been chosen" refers only to that specific Demonic Pact. If you control one and cast another one, you can choose any mode for the second one the first time its ability triggers.

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.