If entities have multiple minis, does that mean that each of them activates separately during the Entity Phase?
> Entities run simultaneously. For example, if a Turn card tells you to resolve BASIC EFFECTS, you should resolve all BASIC EFFECTS (from the currently displayed Entities, but also from the currently displayed Dream Gate tile, etc.) as if they were happening simultaneously. The order of the resolving does not matter because it happens simultaneously in the game's world.
Note that most entities do not perform player targeting actions – this game works a bit differently. It is player characters who check to see if they are within a certain range from the Entity miniature – and if so, something happens to them (the number of Entity miniatures in range mostly doesn't matter).
A situation in which considering the actions of individual Entities separately is actually important to the game is the Entity Movement, described in the "ENTITY MOVEMENT" chapter on page 8 of the manual.
So in the Dogs Entity example, if I have 3 shapeshifter minis around me and the Turn card states a basic action, do I get 3 distress immediately?
> No. As the number of miniatures around you doesn't matter (you have to be in the specific range of at least 1 miniature), you suffer only 1 Distress. In other words, miniatures essentially represent "the single Entity reaching you more easily."