No, the current depends entirely on the voltage applied (and temperature). If you apply 24 volts (to a Samsung F strip) it will pull almost 1 amp of current. It does not matter what's in parallel with it. One strip or ten strips, as long as the voltage does not change, then the current does not change (assuming heat is properly dissipated)
No, they have range of voltage they can be adjusted to - its not load dependent. Once the voltage is set it stays CONSTANT - hence the name "constant voltage" driver. Only when you load it up to its current limit does it become "constant current", and even then the voltage will not go above the setpoint.