First, I do not understand how the LC filter selects the frequency. I know that it's impedance is 'infinity' at the resonant frequency, but I don't understand how it rejects the other frequencies: no matter the impedance of the LC filter for a given frequency, the voltage is applied at the 'X' node and hence every frequency is applied at the input of the diode?
I do not understand the concept of 'shunting a signal to ground'.
You forget that your antenna doesn't have zero impedance (it's not a perfect voltage source, it's more of a pretty imperfect current source): all the current that flows through the LC filter can't flow through the diode – it's simple as that.
Second, I do not understand how the R1/C2 network works? It is said that its purpose is to 'smooth' the input signal in order to get only its envelope at the amplifier input.
Ah, ok, whoever said that mixed a few things up.
Again, C2 shunts high frequencies to ground (makes sense, right? for high frequencies, the capacitor becomes nearly a short), and for that to reduce the voltage that the input of the amplifier sees, there needs to be a source impedance (a series resistance). That's why a RC lowpass filter looks like
input voltage >----| R | --+------> output voltage
Probably whoever claimed R1 was involved in the smoothing thought it was the R in that RC filter network. It's not, because it's not in series, but in parallel to the C. The role of the R in that filter is taken by the voltage drop over the diode.
The R in the circuit above probably just serves to bias the voltage at the output of the diode around ground.
How does it works? Because the output signal of the diode (the AM signal of the selected frequency) is directly applied at the input of the amplifier, so I don't see how the R1/C2 network can smooth anything.
As said, current through the diode leads to voltage drop over the diode. High frequency voltages over a capacitor lead to high current through the capacitor. So, this attenuates/suppresses high frequency voltages.