ESD mats & motherboard safety
A) If I were to put a completely new motherboard straight from its retail package onto the ESD mat, would there be any risk of said mat draining CMOS battery or shorting out RTC circuit powered by CMOS battery? Why? Why not?


The mat typically has a resistance > 1 M&Omega; and &lt; 1 G&Omega;. Such  huge resistance won't short or (really) drain any battery.


  B) Does the situation change if the motherboard have been on for some time before turning it off and immediately putting it on ESD mat? Why? Why not?


No. Same reason as in A. The resistance is huge enough not to cause damaging currents anywhere on the motherboard.


  C) Is it safe to turn the motherboard on while laying on the ESD mat? Why? Why not?


Yes. Assuming the power supply providing the voltages to the motherboard is safe, the voltages on the motherboard itself (12V, 5V and 3.3V) are SELV, Safety Extra Low Voltage.

Important thing regarding A and B:
If you don't wear a ESD strap yourself (which has the same connection as the mat), you might damage the motherboard by the act of putting it on the ESD mat or, when it is already on the mat, by the act of touching it.
The second action is most easy to understand. There can be an Electrostatic Discharge (ESD) from your fingers through the air to the motherboard which drains away even better due to the ESD mat.

Image source  

When holding the motherboard without protection and approaching the map (to put it on the mat), this same ESD pulse can go through your fingers, through the motherboard, through the air, through the mat, which may have the same catastrophic result.