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Protoss become very powerful in the late game, especially on island maps. Killer combos like carriers/corsairs, reavers/dragoons, etc. give much problems to terran and especially zerg players. The best solution for defending this kind of attacks would be psi-storm - and that's protoss again.
Dark Templar Drop
The faster dark templar drop would be possible in mid game, but with sacrificing too much economy. My recommendation for a build order would be:
1. Get an early cyberetics bay and assimilator for mass dragoons,
2. Get mass dragoons :)
3. Expand, then build technology needed for the dark templars.
Of course you won't be stupid to send your shuttle filed with dark templars into unexplored territory, so a few packs of observers can be very handy. In most of the cases, if you drop 4 dark templars, hit the base first as you'll be able to destroy it very quickly and your opponent will be in trouble then. Against terrans you can hit tanks/bunkers if there aren't any turrets nearby, against protoss, hit cannons if there are 2-3 of them and against zerg try destroying as much buildings as you can as you'll be quickly detected. Note that even if you are detected, the dark templars are still a powerful units that can quickly dispatch mass zerglings or marines.
Most of the players think that dragoons are unsuitable for recalling, but it's just the opposite - if you place all of them correctly, you'll be able to recall more than 20 of them. That is equivalent of 40 hydralisks or 60 marines (give or take a few). Having the arbiter close to the recall point, all of your dragoons will be cloaked, so killing detectors is a wise opinion. When recalling, be sure to put some observers among the dragoons for detection (also, most of the people think that arbiters are detectors...).
Carriers are very good units when used in combination with corsairs. With corsair web you'll be able to neutralize most of the anti air units, while neutron flares with interceptors eats up air units. Most of the people cannon up in the early game when going for mass carriers and most of them do it wrong (look and cannoning-up for more information). The most important thing for carrier mass production is an early expansion. The first thing in mind then would be how to cannon up two bases. The answer is - don't. Instead get 2 or 3 cannons in each base for detection and build dragoons for mobile anti air defence. Another important thing are the upgrades - as soon as your cybernetics core pops up, start researching air weapons and armours, while upgrading the shields in the forge. The other upgrades you'll need is corsair web and interceptor upgrade (8 per carrier), and if you want you can get argus jewel for increased corsair energy. Usually attack when you have about 12 corsairs and 6 carriers.