[Bevo]

The question is…. Is the AC on?

Yes, it's on. When you push auto the snowflake doesn't light up because auto is also for heat. You can manually turn on the A/C by pushing the snowflake. If the snowflake is lit, when you push it again the A/C compressor shuts off but the fan keeps the level it was set at.

FYI, all the manuals are available online: INFINITI Manuals and Guides | INFINITI USA

 

[Bevo]

Auto was off

Your picture shows auto is ON. That's what the amber light above the "Auto Recirc" indicates. If the ambient temps are hot and very cold air is coming from the vents, the A/C is either on Auto or you've pushed the snowflake.

If Auto is engaged and you change the fan speed, the Auto indicator on the left column will turn off, but the A/C will still be on. Full Auto controls how the air is blended (hot, warm, cool, cold) and the fan speed.

 

[Bevo]

Do you know if the A/C compressor is always on when in 'Auto' mode?

Compressor is on when you select defrost/defog if the ambient temp is above 23F.
It's explained in the owner's manual. I even provided the link for the online manuals above.