When a psion manifests a quickened power in a round, does this count towards the amount of power points they're allowed to spend per their manifester level that round, or can they conjure the quickened power to the maximum efficiency of their manifester level, than conjure a second power as their standard action to the maximum efficiency of their manifester level?
Power points per round, not per action. You cannot spend more power points per turn than your manifester level. If you spend ten toward your quickened power and you are a 15th level manifester, your next power can only use 5 power points

Also, can a psion quicken a dimension door power to get out of the way of a fireball? If they can, can it only be them, or can they take someone through the door with them per the description of dimension door in that short of time?
A quickened power is still used on your initiative, so no, unless you go before the caster of the fireball.

If the psion readied an action trigger by the casting of a spell by the opponent, then I believe this is possible.

If you go before the caster of the fireball, but the person you want to take with you through the Dimension Door does not, can you take them with you through the Dimension Door when a fireball is coming towards the both of you (provided they're in the space next to you)? Also, do you have to call that you are readying a quickened spell on your iniative in case a fireball comes at you, or can the player dodge the fireball ahead of time, so long as their iniative is before the caster's, without readying this action to do so?

Sorry if that sounded complicated. ;-)

If I had a player get out of the way of a fireball before it was cast, I've have the caster aim at the new location. I imagine if an action is readied, a quickened action is irrelevant, as the readied action goes off first anyway. That seems a bit more like what you're talking about.

It is my understanding that quickened powers still count against their total available per round.

Quickened Actions are Swift Actions, not Free Actions, so they only take place on your round.

You can ready a swift action/quickened spell if your iniative is before the fireball, right?

Not necessarily "when the spellcaster lobs a fireball", since you don't know what the spellcaster is doing, but you could ready a spell to cast Dimension Door once someone starts casting a spell.

The way actions work, it wouldn't even have to be Quickened because your place in initiative would come just before the spellcasters.

Of course, as a DM, I would rule that you couldn't prepare such an action as it has no affect on the spellcaster's casting.