-Does the mm size of the LEDs have any effect on brightness?
Look at the datasheet and note the luminous intensity figures.
The viewing angle is also important, some LEDs are very bright but have a very narrow beam width and appear unlit unless you're looking directly at them.
-Would I be able to use one battery per cube to save money, by running about 3 separate reed switch/led loops to the same battery?
That depends on how powerful the battery is.
-I am thinking I could use some of the magnetic strips you can get at walmart for the magnets, would that work?
That depends on whether the magets are strong enough to activate the reed switches or hall effect seonsors.