What is the purpose of these diodes in half bridge
'Shoot through' is a rather descriptive term for what happens in a bridge when one set of devices turns on before the other set of devices have turned off. A large spike of current flows through the bridge. At best it reduces efficiency and generates EMI, at worst it overheats or destroys the FETs. 

The simplest scheme to avoid this is to use a resistor in series with the gate, to slow down the FET turn on (R1, R5, R8, R9 etc), but then bypass it with a diode, to speed up turn off (D3, D5 etc).