Did Christ have the freedom to choose to avoid crucifixion?

In the same way that all of us have the free will to disobey commands from the Father, Jesus had that same opportunity. I think your police office analogy is good: if the office fails to make the arrest, disciplinary action follows. That's the same in Jesus' case: if Jesus had chosen to avoid crucifixion, there would have been dire consequences: we would receive the full punishment that our sins merit, and Jesus would suffer in the knowledge that we, his loved ones, were lost to him.

The prayer in the Garden of Gethsemane indicates that if there was any other way of achieving our salvation other than betrayal and crucifixion, Jesus would have taken it.

There is a counter-argument here though, and we stray here into a discussion about predestination. Jesus came to fulfill a prophecy. It was clear from what the angels said to Mary, Joseph and the shepherds, and indeed to John the Baptist's parents, and from other events through Jesus' life, that he was the Messiah. The whole point of his earthly life was to lead us to salvation, and so (since there was no other way, as just mentioned) his crucifixion had to happen. So Jesus had a choice, but it was inevitable that he would choose to save us.

I'm not sure that I want to open too much of a wormcan here (maybe too late), but some of this depends on how we define "free will." If we consider it to mean the unrestricted freedom to act on our desires in as much as we have the ability to do so, then Jesus had complete free will...as do we all, generally speaking. (well, there might be times when our freedom might be restricted by outside forces, but those are more environmental than philosophical, and I guess those could very well be considered as part ot the "ability" part of my definition...e.g. if there's something heavy sitting in my lap, then that's what's denying me my "free will" to go the the bathroom; not God).

There is some caveat here that I think is sometimes missed in that, assuming my definition is accurate, then with complete and perfect freedom (and ability), people will freely act necessarily in perfect accordance to their nature. I don't like olives, so even though I have the freedom (generally) to eat olives (i.e. they're freely available to me), what's more important to me is that I also have the freedom not to do so. So while, theoretically and hypothetically, I could choose to eat olives, I won't (it goes against my character), so in the event that I do eat an olive, it actually doesn't support my "free will," but would rather be evidence to suggest that my "will" is not so free after all.

I think this is significant, because it places reasonable limits to avoid potential absurdity with the hypothetical element of "what could he (theoretically) choose..." to the more practical element of "what can/will he reasonably choose if given the freedom to do so..."

To take, perhaps, a silly example. In scripture, God tells us He does not lie. Assuming that He said so in good faith, then we could ask: well, does He have the "free will," to lie? At first glance, it might be tempting to say "yes," because God is omnipotent, and has complete freedom, but if we understand that question to be synonymous with "Could He lie if He wanted to?," then it becomes a loaded question [it's explicitly based on a questionable premise ("if He wanted to")], which makes the question become somewhat inane, because in revealing His character, He's told us He does not want to. The practical answer, then, would seem, to be "no," He does't have the "free will" to lie, because he doesn't have the "will" to lie....IOW, what's actually been asked in this case is "Does God have the free will to act contray to His nature?" but -- assuming again that my definition of free will is reasonable -- this is a nonsense, contradictory question (i.e. "Does God have [the freedom to act in accordance with His nature] in a manner contrary to His nature?")

That being said, I would say that the Bible does affirm that Jesus did have the complete unbridled freedom (e.g. He could have overcome any sort of opposing forces with a legion of angels) to act according to his character, which was destined (from before the foundation of the world) to die on the cross for the sins of his people. He did so freely, but there really was never any uncertainty that he would (as it was planned and determined before the foundation of the world). I think what Jesus is actually highlighting here is not so much that He could choose not to do this (though, theoretically, I guess, it could be argued he could), but, rather, that this was all happening according to His plan and on His terms. The divine plan was playing out here; he was not being killed against his will as it might seem.

ETA: TL;DR version: Jesus was free and able to avoid execution (i.e. he was not foricbly killed against his will), but to ask if he had the "free will" to avoid it seems a contradictory question by my understanding of "free will," because he revealed his "free will" to do so in the fact that he did so and did so freely.

He did have a choice, since He is God. Since all things were created by Him, He is God the Father. As God, He has all authority in the same manner that God the Father does.

As God, He chose to lay down His life on the cross before the creation of the universe. He knew that we would rebel against Him, and that His sacrifice would be necessary for our redemption.

To answer the last part of the question, a good explanation for why He prayed to the Father, and why He spoke of "His Father" being greater than Him is answered here.