Can you replace just the relay with the LED + resistor, leave the diode in place, and re-populate the table ? See if the LED lights up.
Supply: 5V, LED + 330 ohm resistor, no diode, no relay:
Pin state: high, drain: 5V, gate: 5V, LED: off.
Pin state: low, drain: 1.6V, gate 0V, LED: on.
Supply: 5V, LED + 330 ohm resistor is in parallel with the diode, no relay:
Pin state: high, drain: 5V, gate: 5V, LED: off.
Pin state: low, drain: 1.6V, gate: 0V, LED: on.
Supply 5V, LED + 330 ohm resistor is in parallel with the relay, no diode:
Pin state: high, drain: 5V, gate: 5V, LED: off.
Pin state: low, drain: 3.2V, gate: 0.1V, LED: off.
Equally i would see about measuring the resistance of your relay coil, just to make sure its not open circuit or something silly,
Resistance of the relay coil: 55 ohm.
I am really confused. I have double checked the datasheet for the MOSFET and it is wired correctly.
@alsetalokin4017, I'll upload a photo of the setup in half an hour. Let me try to supply 9V to the gate.