Some questions about antistatic wrist strap

should the alligator clip of the wrap always be grounded to the mains earth in a way?

Yes, the alligator clip should be earthed. As should the work surface beneath the work piece.

Is this type of wearing antistatic wrist strap aiming to protect the human from ESD shock or sensitive electronic circuits in general?

It's meant to protect the electronics, not the worker.

Is this wrap also used to protect from 230V AC line hazards or used only for ESD?

The wrist strap does nothing to protect from line hazard, for the user or the work piece.

There should be a 1 megohm resistor between the wrist strap and the connecting wire, so as not to increase the risk to the user if they contact a powered part of the work piece.