Yes it does. You set it using the "Mode" button which is beside the Power button. To set a CD to repeat you need to (1) put the CD in the player; (2) turn the power on; (3) press the Mode button twice (you will see a small curved arrow on the right side of the LCD display); (4) then press the Play arrow. That is all there is to it. The CD will continue to play until you stop it. You can find these instructions in the Operating Instructions under the heading "Selecting a CD Play Mode". If you don't have the instructions you can download them from https://docs.sony.com/release/CFDS05.pdf
The answer is "YES." but you can record only on a cassette tape. This unit will NOT burn (or record) a CD disc. All you need do is point the dial-face front of the Sony Radio towards the piano from any direction which gets the best music sound. Simply turn on the power by pressing the main power button, press the tape source button on the front of the Sony unit. Then, using the controls at the tape deck, open the tape deck cover lid, insert a recordable tape, close the tape lid, then press the record button (that's the button with the red dot) and have your kid start playing. The unit has a built-in recording microphone to pick up the sound. You will also note there is a female port for an audio insert if you want to record from some other mechanical source which includes the bayonet insert.