12V requires quite a lot of amps for enough light, so low DC is not optimal. Also LEDs are current driven devices, i.e. they will be sensitive to voltage changes (even with a current limiting resistor)
Low-voltage doesn't necessarily mean 12V. I think it's anything below about 50, although lighting systems currently marketed as "low voltage" are usually 12 or 24 volts.
The constant current thing is true, but that's not a terribly difficult problem.
