Why is the 240VAC-12VDC giving both 12VDC and 26VAC as output?
From the photos, this is a very simple inverter power supply. The transformer outputs some 9VAC at a multi-kHz frequency, which is then rectified through the two diodes on the bottom left of the first photo, then filtered through the caps on the bottom.

Because the frequency of the transformer output is in the kHz range, the ripple also is. Your multimeter cannot handle this. The AC reading has no meaning.