yarko:
The strip lighting says that each light can be controlled individually; could something like a TLC5940 be used to control 16 of them?
No, they don't work like that.
Each LED already has a controller chip inside it ("WS2811"). You communicate with the string/matrix (using a single Arduino pin) to set the color of each LED.
You can also chain strips together and control them all like one long strip.