Caps = Crap
Fixing your dimming issue - do to list in order:
1. Do the Big 3
2. Get a HO Alt
3. Get a deep cycle battery
As for your system not running for long with the car off. The only solution is to put multiple batteries in your vehicle.
That depends on how the speakers are wired inside the box.
More than likely if you only use one of the "ports" then you'll only be connecting 1 of the subs.
In other words you may need to pull the subs out and see how they're wired in the box and correct it. Otherwise YES wire it exactly how...
Pretty sure that's a bass boost. You can set it at either, but 12 is probably better.
80 will probably be ok on the lpf.
Neither of those setting should effect your amp going into protection.
You should be able to push those subs with that amp with no problem.
I'm guessing you've got it wired incorrectly @ 1 or 2 ohms at the amp possibly bridged. Pretty sure that amp is only 4 ohm stable when bridged and 2 ohm when wired @ stereo.