While recording to chart paper, can I also record the data on a PC via communications?

Yes.
For simultaneous recording, you must add the Ethernet communication interface option (suffix code /C7) to the µR20000. You will also need to install DAQLOGGER (model WX101) and Gate-µR (model WX1) on the PC. With the µR20000 and PC connected via Ethernet, you can use the above software to record data on a PC.