The role of doubt in Zen Buddhist practice?

Case 28 of the Mumonkan
Ryûtan Blows Out the Candle

Tokusan asked Ryûtan about Zen far into the night.
At last Ryûtan said, "The night is late. Why don't you retire?"
Tokusan made his bows and lifted the blinds to withdraw, but he was met by darkness. Turning back to Ryûtan, he said, "It is dark outside."
Ryûtan lit a paper candle and handed it to him. Tokusan was about to take it when Ryûtan blew it out. At this, all of a sudden, Tokusan went through a deep experience and made bows.
Ryûtan said, "What sort of realization do you have?"
"From now on," said Tokusan, "I will not doubt the words of an old oshõ who is renowned everywhere under the sun."
The next day Ryûtan ascended the rostrum and said, "I see a fellow among you. His fangs are like the sword tree. His mouth is like a blood bowl. Strike him with a stick, and he won't turn his head to look at you. Someday or other, he will climb the highest of the peaks and establish our Way there."
Tokusan brought his notes on the Diamond Sutra to the front of the hall, pointed to them with a torch, and said, "Even though you have exhausted the abtruse doctrines, it is like placing a hair in a vast space. Even though you have learned all the secrets of the world, it is like a drop of water dripped on the great ocean."
And he burned all his notes.
Then, making bows, he took his leave of his teacher.

Faith and doubt are dependent on each other. The one is useless without the other. I don't know if you've ever had this experience before, but how fresh and significant is the landscape when you are utterly lost? Everything appears as it is. You don't see that cedar tree where that raccoon was last spring. You don't see that hollow where your brother fell in the river. Instead, every rill in the landscape, every tree, every rock, stands out in bold relief revealing something like its true nature. It's fresh and undistorted by memory and preconception. This is beginners mind, but there's also this visceral, existential doubt, right? I mean, we're lost for godsakes. We have to find our way out! Our very lives are on the line! How are we going to do that when we have no idea where we are and no idea where we're going? This is the great doubt of Zen. Within it, we are lost, exposed, and utterly without direction.

Case 5 of the Mumokan
Kyõgen's "Man up in a Tree"

Kyõgen Oshõ said, "It is like a man up in a tree hanging from a branch with his mouth; his hands grasp no bough, his feet rest on no limb.

Someone appears under the tree and asks him, 'What is the meaning of Bodhidharma's coming from the West?' If he does not answer, he fails to respond to the question. If he does answer, he will lose his life.

What would you do in such a situation?"

Mumon's Comment

Even if your eloquence flows like a river, it is of no avail. Though you can expound the whole of Buddhist literature, it is of no use. If you solve this problem, you will give life to the way that has been dead until this moment and destroy the way that has been alive up to now.

Otherwise you must wait for Maitreya Buddha and ask him.

Mumon's Verse

Kyõgen is truly thoughtless;
His vice and poison are endless.
He stops up the mouths of the monks,
And devil's eyes sprout from their bodies.

Getting lost is not fun. No one likes being in a state of uncertainty. We like having something to lean on whether it's our map, the sutras, etc. When we lose that safety blanket, we lose our desire to go on. When we're lost, great faith is picking a direction no matter how wrong it may seem. Great faith is trusting that we'll eventually figure things out. Great faith is letting go, even if doing so may destroy us. Great faith is the decision to trudge headlong into doubt.

In reverse order:

"wouldn't faith and doubt contradict each other unless they occur separately at different times"

...or unless they are directed to different things: Great faith [on the approach], great doubt [about Truth]. Great resolve: the act of persevering on the approach to penetrate the Truth.

"I can see how faith and resolve would be essential but I don't see why doubt would be?"

Great doubt can be seen as the catalyzer that creates motivation to make one go through great lengths to reach something so hard to reach. It can also be seen as "The Doubt among the doubts", a fundamental, deep and honest question, greater than any other, that no superficial answer would satisfy it, settle it, or make it rest.

This has been explained by Yashutani Roshi in his 10th lecture. The relevance of doubt (Dai-gidan among faith aka dai-shinkon & strong resolution aka dai-funshi) has been explained by him as follows -

The second indispensable quality is a feeling of strong doubt. Not a simple doubt, mind you, but a "doubt-mass" and this inevitably stems from strong faith. It is a doubt as to why we and the world should appear so imperfect, so full of anxiety, strife, and suffering, when in fact our deep faith tells us exactly the opposite is true. It is a doubt which leaves us no rest. It is as though we knew perfectly well we were millionaires and yet inexplicably found ourselves in dire need without a penny in our pockets. Strong doubt, therefore, exists in proportion to strong faith.
I can illustrate this state of mind with a simple example. Take a man who has been sitting smoking and suddenly finds that the pipe which was in his hand a moment before has disappeared. He begins to search for it in the complete certainty of finding it. It was there a moment ago, no one has been near, it cannot have disappeared. The longer he fails to find it, the greater the energy and determination with which he hunts for it. - Page 81, The three pillars of Zen.