1: I'm not sure. Wikipedia and Gamefaqs alike seem to be missing Quickening sections. I can't tell if I'm supposed to be putting in some sort of button combo or not... if I am, I don't see it on the screen. I know you can link other characters' Quickenings to create combos, so... maybe that's what the timer is for? Maybe Andromeda knows more.


2: No, it can't be turned off permanently, and no, I didn't come across anything too special and I covered the whole ship. If you head out the top of the lower level first, you can go to the control room. Take out six guards and you have access to the first terminal -- meaning you can turn the alarm off prior to going back to the storeroom. Then run as quick as you can back downstairs and then up to the storeroom. The alarm should sound just before you get there, but only for about ten seconds. When you get in the storeroom, turn off the alarm from one of the panels (doesn't matter which) and then just run upstairs as fast as you can and toward the exit.

I know you mentioned something in the other thread about needing some levels. My advice to you to gain a couple of levels would be to sit just outside the room with the save crystal, and let guards come to you. Fight until you're almost dead, turn back into the other room, save, and repeat. It will get you the experience points you need to level up without risking a game over.



edit:
re: #1:
How they work
Gaining your first Quickening will bring one new change, you will notice that when your MP is full it will start to glow. Your second Quickening will double your MP as well as splitting your MP into two. The third will triple your [original] MP and will split your MP bar further into three parts.

The split in the MP's are called "Mist Cart"'s short for Mist Cartridges, and the Quickenings use certain amounts of Mist Carts to be used. The first Quickening, uses one Cart, second uses two, and the third uses three. Using Quickenings will drain the whole cartridge, meaning that a level 3 Quickening will leave you with 0 MP and so it should ONLY be used in extreme emergencies (preferably near the end of the battles).

How do I do Combo's
When you select a Quickening you will notice on the bottom right hand corner there will be the names of your three characters, and possibly a button (Square, Triangle, Circle, X) next to their names. Pressing the button will make that character carry out that particular Quickening themselves. If there's no buttons next to the names, press the L2 button to shuffle the list until the button symbol appears. To make matters worse, as the combo's go up, the time limit you have to choose the next combo goes down.
SOURCE

I'm pretty sure it's R2 rather than L2.

My suspicion was correct though. There are no buttons to press to actually perform the Quickening; rather, there are options to chain Quickenings together.