Does Espeon's "Solar Revelation" Ability prevent moving or placing damage counters?
In the steps of resolving an Attack, moving or "placing Damage Counters" counts as an Effect of the attack; it is not part of "doing damage". Therefore, as an Effect it would be prevented by Solar Revelation.
Does Espeon's "Solar Revelation" Ability prevent the flipping of coins for an attack that does damage times the number of heads flipped?
No, it does not. Flipping coins in this context is used to calculate how much damage is to be done. Since "doing damage" is not prevented, the flipping is not prevented either.
If Cinccino with the "Smooth Coat" ability also has "Rocky Helmet" attached to it, can I choose to activate Rocky Helmet first, and then flip for Smooth Coat?
You would need to know whether Cinccino takes damage or not in order for Rocky Helmet to work, so in this scenario you would have to flip for Smooth Coat first.
If I attach a Rainbow Energy to Shuckle with "Fermenting Liquid" Poke-BODY that has 5 damage counters on it, would it be knocked out prior to its Poke-BODY being applied or would I get to draw the card before it's knocked out?
It is a Poke-BODY that is automatically triggered, so you would get to draw the card.
Say a player calls Magnezone's "Magnetic Draw" Poke-POWER, but then realizes they already have six or more cards in their hand. Does that mean his "Magnetic Draw" Poke-POWER is used up, or can he use it later in the turn since nothing happened?
Magnetic Draw says "you may draw cards until you have 6 cards in your hand"; if you have too many cards you cannot "draw zero" so the Magnetic Draw Poke-POWER is not used up.