Cure III is only good in primal fights. It is useless everywhere else, unless you find yourself in a situation where your party members are actually stacked together (never except in primal fights).
Never used a Cure III proc, because it never procs in situations that require it.
You will need to use Cure II much more in endgame fights. The way I determine which button to press is between 80%-100% of a tank's hp bar (and if there's another healer), use Cure; if the tank has fallen below 80% or you anticipate a big damage strike coming soon, cast Cure II. Medica is for mopping up the party damage. When you hit 50, you will get Medica II which is Regen For The Party, which mops up more damage.
If there is a bard in the party, you can afford being a little more liberal with your mp, but automatically assuming the bard should support you is not optimal. If the bard supports you, he/she gets a 20% damage debuff, so it's better if you juggle your own mp to the best of your ability.