The dog park is kind of a mini-location in the fourth dungeon, and is accessed in the "CHOOSE YOUR DESTINY" part of it. The choose your destiny part is a series of connected rooms that branch off into two other rooms each time, so you just pick a path and encounter a series of events. Red rooms have teleporters to one of two locations, one of which being the dog park. The second red room will take you there. Inside there are two events, one with Mint and Chocolate, in which Chocolate asks if you want to fuck Mint, who is in a bitchsuit. The other is a fight with a wolf girl (who has a penis), which results in "mating time" if you lose. Losing to Katie in the choose your destiny section of the dungeon can also take you to the dog park, with her putting you into a bitchsuit before taking you for a walk. (There's two scenes for losing to her, so if you don't get the bitchsuit scene the first time you just need to lose again.)
As for the whisp thing, what she tells you in game is all the explanation you get for now.