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 devence value of her card and items>.

If your attack value is lowered to zero by Dodger, does the hero you are targeting still have to discard to block?

Check the second to last sentence in the Defense action section in the rulebook.

A hero performs an attack with a red attack card, but the final attack total is 0 after modifiers. Does the defending hero still have to discard a card to defend, or can they choose not to defend and simply survive because the attack total (0) is not higher than the defense total (0)? The rulebook says that if you don't defend you are defeated, but the detailed defense rules resolve defeat by comparing attack vs defense totals. Is discarding to defend mandatory for any attack, or only relevant when the attack total could actually defeat the hero?

It's in the rulebook in the Attack action. Page 14.

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”