Liquid / water level detection safely using AC voltage - Industrial circuit explanation
You have quite a few questions. 


No the cap across the transformer secondary does not short it, maybe it reduces some higher frequency transients from external sources or from the diodes switching. Pretty sure it will work about the same without it. 
The current through the probes is AC (ignoring a brief bit when first turned on) so long as the series capacitors on pickup probes E1/E2 are not damaged. On the positive half cycle the current goes through the M7 diode (and electrolytic cap etc.), on the negative half cycle it goes through the 56K resistor. 
24VAC is generally regarded as safe and there is series impedance. Some electrolytic action may take place during the half cycles, but it's probably not that important. Keep in mind that some water is quite high in resitivity so a higher voltage allows more reasonable impedances to be used in the detection circuit for a given probe design. 
This question is beyond the scope of this forum, I think. I have a commercial cappuccino machine that uses a similar method to monitor the water tank, so it was approved in Italy. The probe metal(s) would have to be safe for potable water contact, at least.