Can I record my performances?

Yes, by using the Recording feature in your application.

[WeDJ for iPhone, WeDJ for Android]
Tap the record button (light red-circle icon) on the screen to start/stop recording.
You can check the recorded sound by tapping the area where track titles are displayed at the top of the screen and select from [REC LIST] at the bottom right.

[djay]
Tap djay icon > [REC] on the top of the screen and tap [Start New Audio Recording] to start recording.
Tap [Stop Recording] in the same procedure to stop recording.
You can check the recorded sound by tapping [Recording 201..] on the screen.

[edjing Mix for iOS]
Tap [・REC] on the top of the screen to start/stop recording.
You can check the recorded sound by tapping the note icon on the screen and tapping [My sets]>[My recordings] on the bottom.

[edjing Mix for Android]
Tap [・REC] on the top of the screen to start/stop recording.
You can check the recorded sound by tapping the note icon on the screen and selecting [MY MIXES].

[rekordbox]
Click the [REC] checkbox on the top of the rekordbox screen. The Recording panel will be displayed in the middle of the screen.
Click the REC button in the panel to start/stop recording.
When recording stops, the recorded file is saved as a WAV file.

For details, refer to "Recording" (PERFORMANCE mode) in the rekordbox Operating Instructions.
You can read the rekordbox Operating Instructions [here].