What is the use of an op-amp whose output and inverting input are connected to ground?
The first OP-amp is actually creating the circuit ground. The 7810 creates a stable 10 volt, which is then divided by the voltage divider R2 and R3, filtered by C3 to make a stable 5 volt level relative to the most negative level.

The OP-amp then buffers this, and the rest of the circuit uses its output as the reference ground. Remember that ground in a circuit like this is just a convenience, a node that is used when referring to other voltages.