I've had the issue of slow detection of projectors. Sometimes the sequence of connecting makes a difference.
Try to set everything else up first, and hold connecting the adapter to the MBA for the last thing.
i.e. turn on projector, connect the HDMI cable to the adapter, connect the cable to the projector, select HDMI on the projector.
Once everything else is set up, THEN connect the adapter to the Thunderbolt/mini display port.
See if that sequence will work.
Apple was aware of this issue. On the Macbook Air mid-2012 and Macbook Pro 13" mid-2012, the issue was resolved (or was intended to resolve) that issue. However, it doesn't look like they're aware that older Macbook Airs are affected. I would consider contacting Applecare just so they're aware of it.