Confusion with interpreting a functional safety diagram of a motor driver
1-) Is my understanding of the mechanism correct?

You have not made your understanding clear.

2-) What does "Power ON" switch next to the K1 represent in the original diagram?

It appears that the external "Power On" switch applies 24 VDC from an external supply to the coil of an external relay or contactor that connects mains power to the controller. There may be some internal contact between X8/1 and X8/2. That contact may close only after some internal evaluation of readiness to receive mains power.

3-) How should I interpret relay output ratings above: 30V DC 1A and 42V AC 1A? Are they the ratings for the relay K1?

Those are the contact ratings for internal relay contacts that open or close to indicate controller operating or fault conditions.

4-) The datasheet does not mention in which conditions "Fault Relay" activates. Does that mean it activates when there is signiant amount of current passing through earth? "(terminals X8/1 and X8/2 ) is signaled by a floating relay contact" What do they mean by floating relay contact here? Is that the relay inside the driver?

There are likely a number of fault conditions associated with the fault relay. They should be listed in the manual. The relay is inside the controller (driver). "Floating" means that the contact is connected only to terminals for external use there is no internal connection to power, ground or anything else.