is the easy one. It means "succeed". As such it should not need great explanations.
emphasises the final outcome of a process.
Wie ist seine Entscheidung ausgefallen?
The other two would be wrong here.
This is extremely hard to pin down, but I would say it emphasises the process far more than ausfallen. Although they overlap in the participle, the difference becomes apparent in the basic verb:
Der Junge gerät nach seinem Vater./The boy comes after his father.
This is clearly a process and not just an outcome! The others can't be used here!
We are not concerned with the other meanings of geraten here, but they might help you grasp, why I claim it has a more procedural flavour:
Wie bist du da hinein geraten? / How did you get into that (situation)?
As for your examples
All three can be used, but "ausgefallen" sounds somewhat odd to me. It has a slight flavour to me, as if there were previous trials in which the baker failed to satisfy and this time the cake came out good. But these are definitely slightest nuances.
This is a very good example. "Ergebnis" implies some finality and hence only "ausgefallen" really fits. Well done!
This would suggests, to me at least, something else was desired. For reasons I can't pin down, it sounds a bit unnatural. On the other hand:
Die Haare sind etwas grün geraten./ The hair turned out a bit greenish. ,
would be completely natural.
This sounds very odd, unless you are Saint Peter, making the weather,or talking to him. Note, however, that this holds true for all three choices.