Yes, an RGB controller is included in the set. You don't need to order anything extra unless you plan to install more than six fans. You don't need a "JCORSAIR1" connector, or at least I didn't have one on the mainboard. You just need a free USB port on the mainboard, a SATE power cable and somehow you have to connect the fans to the mainboard. If your mainboard doesn't have three connectors for "FAN_CHA" i.e. chassis fans, I advise you to first find out about so-called 4-pin PWM fans Y-splitters. I use this one
EKWB EK-Cable Y-splitter for 3x 4-pin PWM fans (10cm) but I ordered it directly from the manufacturer because Digitec once again charges an absurd amount of "Schweizhochpreisinsel"-surcharge for it. A cable like this including shipping will cost you less than 10€ on the manufacturer's website (including 7€ shipping) and if you order more stuff at the same time, the shipping costs are rather okay.
But if you need it quickly, you can order it for the higher price at Digitec.
Here is a tutorial for the installation. There you can also see what you need:
https://www.youtube.com/watch