Sierra showing incorrect monitor in Display Preferences
To answer your questions about running a 4K display with a Mid 2012 MacBook Air 11"/13":

Unfortunately the 2012 MBA isn't technically able to do so.

The 2012 MBA has an Intel HD Graphics 4000 GPU which supports only up to 2560x1600 px via its Thunderbolt port directly. Attaching another video adapter (like HDMI) may reduce the maximum possible resolution.

Depending on the external display it may choose a lower resolution if if the source (your MBA) is not capable of driving the display's native resolution. It appears that your MBA reduces the resolution to 1920x1080 px when you connect it to the display's HDMI port.

If possible use a miniDisplayport to Displayport cable to connect your computer to the display. You may be able to use a resolution of 2560x1600 px.