Does God have free will? To clarify the question - when I mention God here I am referring to God the Father and by free will, the best non-contradictory definition according to the Bible I have heard is "the ability to act apart from one's nature". 

When I first came across this question it sounds almost absurd. Of course God has free will, He willed us into being. But maybe willing something and free will are two different things.  And maybe we have a hard time thinking about this because we personify God to have human traits and the ability to act apart from our nature is a human only trait.
I would argue that the premise that free will is "the ability to act apart from one's nature" is essentially flawed.  Free will is the ability to act within your power according to your nature.

However, regardless of your definition of free will, the question is also influenced by a finite viewpoint.  God is both infinite and perfect.  He has free will, and his will is perfect and his nature is infinite such that his will is perfectly executed.  Being both infinite and perfect in nature, it is not possible for any of the three persons of the Trinity to ever act contrary to their perfect nature - that's one of the reasons it's possible for God to be three persons without inter-relational conflict.

We tend to over anthropomorphize God, and we must be very careful not to reconstruct God in our image, instead of the other way around.  We image God but imperfectly; we are like him, but we are not identical to him (as if that needs to be said).

In a mysterious way, within this creation, God has created us sufficiently in his image such that we have free will; and we will according to our nature, which is fallen and corrupted.  Somehow, God has organized things in this dimension (or these 4 dimensions, if you like), such that we can exercise our free will without contravening his.  It's as if he has deliberately chosen to limit himself for the sake of our creation - how this resonates with his will I cannot explain... as I said, it's a mystery.