I have encountered a lot of different theories concerning the Battle of Hastings and Harold's death. Most recently, I read Harold: The Last Anglo-Saxon King by Ian Walker.
Walker claims that most of the things I was taught at school are probably wrong: the Normans did not break the Saxon shieldwall by cleverly faking a rout; Harold did not commit to battle too soon; his army was mostly fresh with only his housecarls having made a rapid march from Stamford Bridge; the Saxons were likely to win the battle... until Harold really was killed by an unlucky arrow to the eye.
Is there any scholarly consensus on this? Is there anything about the battle that we know for certain?
You may find this review of Walker's book interesting: http://www.journals.uchicago.edu/doi/abs/10.2307/2903438.
I've not read Walker's book, but from what I can tell based on your synopsis and the review, it has serious issues. His gravest sins seem to be undue speculation, partisanship, and a willingness to ignore sources that disagree with his theories.
The English accounts of the campaign are weak on details. The Anglo-Saxon chronicle states that:
Then Earl William came from Normandy into Pevensey, on the eve of the Feast of St Michael, and as soon as they were fit, made a castle at Hastings market-town. Then this became known to King Harold and he gathered a great raiding-army, and came against him at the grey apple-tree. And William came upon him by surprise before his people were marshalled. Nevertheless the king fought very hard against him with those men who wanted to support him, and there was a great slaughter on either side. There were killed King Harold, and Earl Leofwine his brother, and Earl Gyrth his brother, and many good men.
So right there, we see that the English themselves thought the battle had begun prematurely. Harold's intention was most likely to shadow William's army, menace his foraging parties, cut off his supplies, and force him to withdraw to the continent. As such, he took up a very strong position with his army concentrated. William rolled the dice and attacked Harold rather than retreat.
Any tactical details of the battle and campaign must be drawn from the principle Norman sources: William of Jumieges, William of Poitiers, and Orderic Vitalis. Based on these, there is strong consensus that the shieldwall did collapse late in the day owing to a combination of exhaustion and attrition. The first Norman rout (rout is putting it a bit strong - the cavalry were repulsed in disorder) was probably genuine, but after that the Normans made repeated feigned attacks to draw the English out. We don't know when - or if - Harold caught an arrow; one account has him being cut down in hand-to-hand combat late in the day.
So yes, it was a close battle, but Norman victory was not reliant on anything so random as a lucky arrow. The English chose ground that was ideal to an infantry army defending against cavalry, but a combination of clever Norman leadership, superior tactical mobility, and English over-eagerness made the difference.
For a detailed but nuanced military account of the campaign, albeit one from the Norman perspective, see John Gillingham's William the Bastard at War: http://deremilitari.org/2014/04/william-the-bastard-at-war/
You might also enjoy this series of posts I made on the subject: