Why the Anet A6 can't do .05 layer height
I'm not familiar with the Anet A6 specifically, but as many other things in a 3D printer, the minimum layer height is co-determined by a number of factors.  For the Z-axis the factors I am aware of are:


The number of steps in the stepper motor
The geometry of the lead screw
The tolerance with which the lead screw has been machined
The microstep settings
The quality of the stepper drivers
The amount of play and flexibility of the X-axis gantry
The ration between filament and nozzle diameter
The precision of the extruder's stepper motor
...


Many printers that claim to have ridiculously low minimum Z-layer height do so by relying on the mathematical model only.  It goes something like this: the lead screw has an offset of 1mm per revolution, the stepper motor makes 200 steps and is set for 32 microsteps per step so the minimum resolution should be...

1mm / (200 * 32) = 0.0002mm


The reality is however different.  For example: the lead screw may have been machined with a tolerance of 0.002mm, so to make sure a layer has a thickness >0mm its heigh should be >0.004mm (20 times the theoretical limit computed with the microsteps).  But to make sure layers have the same height ±10% you would need to increase the minimum layer heigh of an order of magnitude, bringing it to 0.04mm.  A similar reasoning applies for the amount of play in the gantry, while the ratio between filament and nozzle influences the minimum volume of plastic that can be extruded per step (for very thin layers you want to be able to extrude a minimal amount of it).

At the end of the day this is a typical case of "a chain is only as strong as its weakest link": the minimum layer height of a printer is affected much more dramatically by the limitations of the most imprecise component in the printer than by the performance of its best one.

As for your question about breaking the printer by issuing g-code with very thin layers: should you issue gcode that requires layers below that limit, the firmware will simply print at the same z-height (see comments, credits to Tom).

I'm unaware of people having permanently damaged their machines by issuing code with too thin Z-layers.  But given how the firmware operates, I'd expect the quality of the print to be negatively affected, the filament to be possibly chewed by the cobbed wheel and in extreme cases cloggig of the printer head.