if I send 19V in to the current led driver, would I get 19V out in the led-pcbs
No you get what ever voltage that is needed to drive the current through the LEDs. That is what a constant current supply does. It automatically and constantly, adjusts the voltage so that the current is guess what - constant.
Is it possible to use the BC337 NPN transistor
For the feedback transistor yes.
Which power transistor/Mosfet would be suitable?
Lots, it depends on what you can get locally.