should AC ground (earth) and DC ground be connected?
My opinion only. 

Unless there is a regulatory requirement for your circuit ground to be connected to the AC power ground, I would NOT connect them together.  The main reason to keep your circuit floating is to avoid possible ground loops that might occur if you connect your micro controller to another piece of equipment (computer, etc).

There are specific classes of product where there is regulatory requirement that the circuit ground must be connected to AC ground.  But if your product is not one of those, I normally wouldn't do it.