vout/vin isn't zero, it's about -42 at 1 Hz starting frequency according to the plot. At GHz frequencies, vout is shorted by 1 nF Cout. Unreasonable setup, inappropriate plot parameters. You may want to plot with log frequency scale to better see what's happening.
The resistor has no effect during first part of AC analysis, the inverter is effectively operated open loop. By chance, it's biased almost to maximum gain, see DC analysis plots.
Wrong, the gain is limited by the time constant of inverter output impedance and 1 nF Cout, I guess µs range respectively some 100 kHz cut off. You still didn't plot the gain with reasonable frequency scale?