The diode is used to turn the transistor and LED on when the output is low indicating charging in progress I guess. If it was not there then it would interfere with the circuit when in float mode.
The capacitor is to prevent the op-amp from oscillating.
It's only the current which is sensed and this is the 0.2R resistor and a the reference for this comes from the 15R.
This circuit will switch to float charge when the current is below a certain value set but the 0.2R and the 15R.