Can I input recorded sound on DJM-REC to the connected DJ mixer?

Basically, yes.
However, you CANNOT do it when you're using an analog effector and DJ mixer's SEND/RETURN selector switch selects [1/4" JACK].
To do it,
1. Make the DJ mixer's SEND/RETURN selector switch select [USB].
2. Make sure SEND/RETURN ON/OFF button lights up.
3. Tap the SEND/RETURN ON/OFF button.
Please refer to the Operating Instructions of the DJ mixer for the condition that the SEND/RETURN ON/OFF button lights up.