How does a DC-AC inverter's DC input current change with time?
Yes, it's all about right. The 400 volt supply is an ideal DC voltage source, so its output shows absolutely no variation, even though its current has horrendous spikes.

You'll notice that the input current spikes have an approximate sine envelope, but modulated, And you'll notice that the crossover point is not at 0 volts amplitude. This is because of the 5 mH inductor on the output.

The output voltage has "bunches" of voltage spikes which are all positive or negative, and which correspond to the positive and negative divisions of the current waveform. Those spikes are the direct output of the bridge, so of course they are spikes. If you expand the waveform, you'll see that the pulse width of those output spikes corresponds to the amplitude of the current waveform.

What you want to do in order for things to seem to make more sense is increase the 1 ohm load to 10 ohms, and swap it with the inductor, so that one end is now grounded. If you put a scope across the resistor, I think you'll find it closer to your expectations.