Dodger

If Dodger plays Blue I to defend against Emmitt's primary red when Dodger has 2 or more empty spawn points in radius and no other modifiers apply, is Dodger's total 10 or 12?

Simply follow the text on Emmitt's red verbatim: Dodger has to "use the initiative value of her card and items" <instead of defence value of her card and items>.

Dodger's Tier III alt red says to push "up to 2 minions". Can Dodger choose to push only one of two minions in range, or must both be moved?

Correct. Up to 2 means “0, 1 or 2”.