Continuity mistake: After Edmund has been rescued and is talking to Aslan on the cliff, as he starts walking back to the tents, Aslan in the background is suddenly taller than him, even though he is further from the camera. In the next shot Aslan's height is normal again.
Continuity mistake: When Edmund first enters Narnia, he is attacked by the White Witch's henchman, who pushes him into the snow. In some shots, there is a considerable amount of snow in his ear, and in others, its completely gone. This switches several times. (00:29:15)
Continuity mistake: When Lucy, Edmund, Peter and Susan fall through the wardrobe together, as they stand there the amount and pattern of snow on Edmund disappears, then reappears, and changes drastically between shots, while they're arguing before they head off. (00:40:40)
Continuity mistake: When Lucy, Peter, Susan and the Beavers speak with Father Christmas, the position of his large bag when it's on the ground, and then in the sleigh, distinctly differs between shots.
Continuity mistake: As Edmund lays prisoner in the White Witch's cell, the long chain dragged along the floor, binding his feet, changes position during shots. (01:03:35)
Continuity mistake: When Edmund falls backward into the snow, after his near hit by the White Witch's sleigh, his dressing gown changes from hanging down over his knees, to folded back, over his chest. (00:28:45)
Continuity mistake: Peter and Aslan are standing, looking out over Narnia and in frontal views Aslan is only as tall as Peter's shoulders. But from reverse shots Aslan is taller than Peter.
Continuity mistake: When Lucy first enters Narnia, her hair changes between shots from being straight and neat to being messy and slightly curly.
Continuity mistake: When the four Pevensie children are standing on the Coombe Halt platform, waiting, Susan is holding her brown suitcase in one shot, then in the next shot it's on the floor, and she picks it up once again. (00:06:50)
Continuity mistake: When Edmund is sitting in the White Witch's sleigh he is eating the cakes. He switches from eating with his left hand to his right and then back to his left, dependent on the angle of the camera. (00:31:10)
Continuity mistake: As Lucy, Peter, Susan and the Beavers trek through the snow towards Aslan, their shadows repeatedly change in consecutive shots, such as going from long shadows to their right, to hardly any, to long again.
Continuity mistake: During and before the battle between Aslan's and Jadis' armies, the red chin strap on Peter's helmet changes position a few times. (01:43:35)
Continuity mistake: After Edmund falls through the wardrobe, into Narnia, his hair changes from bushy, brushed back and messy, to suddenly flat and combed between shots. (00:27:25)
Continuity mistake: When Lucy first meets Mr. Tumnus, he holds an umbrella and the strap is either over or under his fingers, depending on camera angle.
Continuity mistake: When Lucy discovers the covered wardrobe during hide and seek, the way the cloth is draped over the wardrobe, and its folds, differ in the overhead close-up, just before she opens it.
Continuity mistake: After Susan and Lucy say they don't have fun anymore, they splash each other in the water, before being surprised by Maugrim, and a few moments later, Peter runs into the water to rescue them. The water stains on their clothing noticeably differ between shots.
Continuity mistake: When Edmund sits in the White Witch's sleigh, drinking hot chocolate, the liquid on his lips disappears, then reappears, depending on camera angle. (00:30:40)
Continuity mistake: After Edmund rejoins his siblings, he has a blood-clotted cut on his lip. During the scene where the White Witch comes and meets with Aslan, the cut changes from big, to almost-healed, then back to big.
Continuity mistake: After Lucy gives Mr. Tumnus one of his packages he dropped, he puts it under his arm. The package is rectangular, yet in the next shot we see a long tube shaped package is under his arm as well. (00:14:50)
Continuity mistake: When Lucy first enters Narnia and meets Mr. Tumnus, she hides behind the lamp. When she comes out from behind it she has snow up the front of her dress to her waist. In the next shot she only has snow at the hem of her dress. (00:14:50)
Answer: Spoiler alert: this gives some important plot twists away. Sometimes a bit of unresolved mystery improves a story, and I think this is the case here. But the book partly answers your questions. At the end of the last chapter it is shown that Mrs MacReady thinks the wardrobe is just a piece of furniture. She knows nothing about Narnia. But Professor Kirke amazes Peter, Edmund, Susan and Lucy by expressing familiarity with Narnia and explaining that a wardrobe might well be a portal into Narnia. If C S Lewis had not written any more books after completing "The Lion The Witch And The Wardrobe" Professor Kirke's knowledge of Narnia would probably have been an unresolved mystery. But C S Lewis later wrote "The Magician's Nephew" which tells how Professor Kirke visited Narnia as a boy. The final chapter of this book says he took an apple back with him, which he planted in his garden. It grew into a tree, was cut down and made into the wardrobe. So Professor Kirke was not consciously aware of what the wardrobe could do, but with hindsight, he realised that he had set up a chain of events that caused the children to discover Narnia.