"do I have to ground the metal faucets that connect on the end of plastic piping? Can these anyhow turn out "hot" since they are not connected to metal piping?"

No and no.

"What happens if a hairdryer falls in the water (remember the piping and the bathtub are plastic)? My bet is that the breaker will trip, but the GFCI will not since none of the current "escapes" to the ground (bathtub is basically isolated right?)."

Not correct.

The GFCI will trip. GFCIs operate by sensing a difference between the current coming into the receptacle and the current leaving the receptacle. The GFCI will trip when this difference is about 6 millivolts.

If you are on a grounded electrical system, the equipment grounding conductor of the bathroom circuit grounds the GFCI back to the main panel.

"Or do I worry too much and no harm can happen since the piping is plastic? :)"

You worry too much because you have no idea how GFCIs and how residential wiring works.

Relax.

At the main fuse box, I'm receiving 3 wires that enter the apartment - line (220AC), neutral and ground

Not correct. You have two hot lines at 110/120 each and a combination neutral and ground.

"What happens if a hairdryer falls in the water (remember the piping and the bathtub are plastic)? My bet is that the breaker will trip, but the GFCI will not since none of the current "escapes" to the ground (bathtub is basically isolated right?)."

Not correct.

The GFCI will trip. GFCIs operate by sensing a difference between the current coming into the receptacle and the current leaving the receptacle. The GFCI will trip when this difference is about 6 millivolts.

Maybe nothing since without a grounding (3-wire) plug or some pother path to ground the current will NOT be unbalanced (hot not matching neutral through LOAD side terminals).

Current is measured in amps (amperes) not volts (or millivolts).

To trip GFCI there MUST be another path for the current to take.

No path, no current, no GFCI trip.

If you then touched the water and provided another path (possible not not guaranteed) the GFCI would trip.

That path must be present and be low enough impedance to allow ~6 milliamps to flow in the circuit with the voltage available.

Posting from Croatia !

GFCIO devices work the same everywhere.

"Current is measured in amps (amperes) not volts (or millivolts)."

That is correct. Current is measured in milliamps not millvolts for GFCI trip currents. Sorry for mis-speaking.

"To trip GFCI there MUST be another path for the current to take.

No path, no current, no GFCI trip."

That is not correct. There is no need for an equipment grounding conductor to be present for a GFCI receptacle to trip.

GFCI receptacles trip by sensing a current imbalance between the hot and neutral. This imbalance causes a magnetic field to form in a metal ring through which these conductors pass. It is this magnetic field that causes the breaker to trip in the GFCI device...not the presence or absence of an equipment grounding conductor.

This is why GFCI receptacles can be used to replace ungrounded receptacles on ungrounded circuits.

See Art 406(D)(E)(3)(a) which says when replacing receptacles when no equipment grounding conductor is present that "Grounding-type receptacles supplied through
the ground-fault circuit interrupter shall be marked 'GFCI Protected' and 'No Equipment Ground'."

It is not necessary for an equipment grounding conductor to be present for a GFCI to work.

"To trip GFCI there MUST be another path for the current to take.
No path, no current, no GFCI trip."
"That is not correct. There is no need for an equipment grounding conductor to be present for a GFCI receptacle to trip."

I did not say a "equipment grounding conductor" was required.
I said "there MUST be another path for the current to take."
Without another path all the current will remain on the neutral and there can be no imbalance.
The path could be through metallic piping than is bonded to ground, or any other path that will allow 6 milli-amps to flow with the available voltage, but there MUST be another path for this current to take.

If you grab the hot and neutral of a GFCI protected circuit you can be killed if there is not another path that unbalances the current the GFCI is measuring.

Current does not Âpile up or Âcollect but flows in complete circuits.
The common demo of placing a drop light into a plastic bucket of water ONLY works if the grounding conductor is present in the drop light.
Otherwise there is NOT a path to unbalance the hot-neutral current.
The light stays lit very nicely.
If you touch the water and provide a path the GFCI will trip, but not until that path is created.

" is not necessary for an equipment grounding conductor to be present for a GFCI to work. "

Very true.
A grounding conductor is not required, but ANOTHER CONDUCTING PATH MUST BE PRESENT.

It is painfully easy to have the path, but it must be present.

A plug in GFCI tester does not work on GFCI devices without a grounding conductor.
The tester shunts the current to the ground connection to unbalance the hot-neutral current.
You can push the tester button till the cows come home and nothing happens.
No ground connection, no unbalance, no trip.

This is rather basic electrical stuff.
Without a closed circuit current does not flow.