The Codex makes it pretty clear that asari are the finest warriors in the galaxy (even turians agree), obviously they were simply toned down in-game for balance reasons.
Anyway, the reason I love playing asari is not because they're awesomely powerful, but because I just find them fun to play. They have interesting, mostly active power sets with very little cheese on top - and my favourite dodge in the game. They're not cheap grenade tossers (aside from vanguard), teleporters or wallhack abusers. To be effective with them requires good positioning, good awareness and good tactical decisions.
Asari kits are extremely well-balanced as a whole. None of them are weak, yet none of them are among the most OP kits either. Valkyrie is the only one of them that could even begin to make the case for being the strongest kit in its class.
As for the actual questions:
1. Commando Adept is my favourite kit in the game. Huntress would be second-favourite asari kit.
2. Justicar and Vanguard share the last spot on my asari list.
3. My favourite asari builds:
Asari Adept / Wraith
Asari Huntress / Claymore
Asari Valkyrie / PPR
Asari Vanguard / Harrier
(I still haven't figured out a favourite build for Justicar, I just keep constantly respeccing and changing the loadout.)