Is this 7805-based design correct?
The circuit is wrong.  

A 7805 can be used in an adjustable mode as per the diagram you provided - using R1 and R2, BUT it is not a good regulator to use this way as it draws substantial and variable current via its adj (ground usually) pin which leads to poor regulation. Using an LM317 in this manner is much better - it is designed to be used like this.



The DC voltage will be ~= the peak voltage = 1.414 x VRMS_AC.
 (1.414 = sqrt(2))

So 12 VAC x 1.414 =~ 17V
 17 - 2*x 0.7 = 15.3 volts.
 The 7805 or LM317 can have 2+ volts internal drop leaving about 12.5 to 13 volts possible at the output.

Use an LM317 if possible - datasheet here.
 Fig 5 in the datasheet is the same effectively as above but the reference voltage is 1.25V.  



To protect against the back polarity problem that you mentioned, connect a diode from output to input (usually non conducting). If Vo > Vin then the diode will conduct and protect the regulator.