Double Attack is an extra attack on your turn if you don't move.

Twin Attack is an extra attack every time you make a regular attack, but it has to be against the same target. That means it is as strong as you say- on Komari's turn, she can make four attacks if she doesn't move. She gets a Twin Attack on her "normal" attack, and a Twin Attack on her second "Double" attack. Twin Attack applies in other scenarios as well- e.g. on attacks of opportunity, and immediate "free" attacks received from commander effects such as from Senator Padme Amidala. It's important to remember that the Twin Attack is always against the same character and can't be used on someone else even if your first attack defeats the target. It's also mandatory so you can't choose to not make a Twin Attack (whereas you can choose not to attack again with Double etc).

Lightsaber Assault is not redundant because it allows her move (up to six squares before or after attacking) and still make the same four attacks (it allows two like Double, and then you also get Twin), which you can't do when using Double Attack.

Not having Melee Attack on her card was a misprint which was corrected by an errata from Wizards of the Coast.

TL;DR: Double + Twin means she gets four attacks while not moving, Lightsaber Assault + Twin lets her make four attacks on the move, and she does have Melee Attack.