1. When making an attack using an attack spell such as Force Missile with Ezren, the card says "For your combat check, discard this card to roll your Arcane die...." His Arcane die is his Intelligence, but do I also add the +2 from his Arcane skill to the final result?
Yes. His Arcane die is actually a d12. His Arcane skill consists of that die plus the modifiers. So he gets to add in his Arcane modifier (+2) and since his Arcane skill refers to his Intelligence, he gets to add in Intelligence modifiers.
Here is a longer discussion about all that.
2. And does using Force Missile as an attack count as an "Arcane check" for purposes of Ezren's Evoker power that reads "Add 2 to your Arcane check with the Force...trait(s)."
Yes. X check means that either the skill or trait of X is present in the check.