For high impedance connections (guitar, bass, some acoustic pickups and other) yes, you would need a DI box to interface with CQ inputs – other than the HiZ inputs on the CQ-20B of course!
For lower impedance (generally line level) connections, you can connect directly to the CQ.
Regarding the output socket, this is balanced, but a TS 1/4″ Jack can be used.
So, could the 18t inputs get damaged if you don’t use a DI box?
No, but the instrument won’t sound quite the way that it should. The most noticeable difference, will be a loss of high end content. It is best to properly match the impedance, for whatever you are connecting to your mixer.