It should be the same, but the dropout does change with current passing through, so the more you have in parallel, the less of a drop out for the same given load. However it's not free lunch though, the quiescent current stacks up with each additional regulator (the amount of current the...
As the voltage drops, the current will have to increase to supply the same net power to the LEDs. The LED driver takes care of that till the voltage is below what it can work with and then your light finally goes out.
I'm guessing it's some kinda coincidence, seriously doubt a bad battery can...
1. It's not quite so linear, you have LED efficacy thrown into the fray, as well as human perceived brightness. There was a whole thread dedicated to this. Look here.
http://www.candlepowerforums.com/vb/showthread.php?179765-LED-perceived-brightness-and-efficiency-linear-vs.-PWM-control
2. Not...