Why does the diode current waveform look like this in a full-wave rectifier?
The current should follow sign of the voltage.  If you measure the current after the rectifier, it has already been rectified and so the current will only flow in one direction as the diodes will prevent it from flowing the 'wrong way'.  However, if you measure the current before the rectifier, then it will follow the sign of the input signal.  Since the input signal in this case is a sinewave that has both positive and negative peaks, you will see both positive and negative current flow.  

It looks like you are measuring the current at the input of the bridge rectifier and the voltage at the output of the rectifier.