Do the electrons that charge your phone come from the network and more specifically from the conductors in the power distribution cables?
The answer here is no. You are correct that at the generating station the electrons get their initial push as some other form of energy is converted to electrical energy. The current produced by the generating station is alternating current typically with a frequency of 50 or 60 Hz. This means the flow of electrons is changing direction 100 or 120 times a second. The electrons oscillate back and forth but don't end up moving along the wire at all.
So, there certainly are not any electrons making it from the generating station to your phone but in fact there aren't even any getting from the cables in your home to the phone. This is because of what happens in your phone charger. To step down the voltage of your home electrical outlet (typically 120 V or 220 V) to about 5 V needed by your phone a transformer is used. The transformer has one coil of wire connected to the power network and one coil connected (indirectly) to your phone. There is no conducting path from one coil to the other.
Is my phone getting charged from electrons of the metals?
Keep in mind that when you charge your phone there are just as many electrons leaving it as entering it. It's not getting charged up in the same sense as when you rub a balloon and give it a static charge. We say "charge the battery" but in fact the flow of electrons is used to store energy in the phone's battery by driving a chemical reaction. As you use the phone a reverse chemical reaction occurs and energy stored in the battery is used to push electrons around in the phone.
And if they do, what's the speed of the electron flow?
It turns out the electrons flow surprisingly slowly. The Drift velocity page on Wikipedia has a sample calculation. With 1 amp of current flowing in a 2 mm diameter copper wire the drift velocity is just 0.000 023 m/s.
