The balloon is not a hot air balloon, but a hydrogen balloon. Such balloons were the only practical balloons of this period, before compact propane fuels were available. They are recognisable by being smaller and spherical, without the lower opening of a hot air balloon. And of course, they don't need a burner.
In the opening scene, Oscar (who is "Oz") is trying to impress his new assistant by offering her a music box owned by his deceased grandmother, "a decorated war hero", a seriously embarrassing English grammatical mistake that no American male would have made when speaking to a female in 1905, for at that time, everyone knew the difference between "hero" and "heroine", and there was no such thing as "political correctness".
When Oz first meets Theordora, his bag disappears from his hands then reappears twice. The first time is when the fairy spits on him and he covers his face, the second is when he conjures the magic flowers.
When Oz is talking to Glinda about leaving and packing his bag, he puts the deck of cards into his bag and the table is mostly empty. Next shot the cards are laying on the table again. Finally, he picks the deck up for a second time and places them in the bag.
Throughout the movie, the facial hair thickness of the Oz was not as it was at the beginning shots of the movie. The thickness of the hair slightly changed between short to shorter, then back to normal.
When Theodora and Evanora are on the city wall watching the dummies approach, the broom in Theodora's hand flips over between shots. When facing out, the bristles are down, but when she turns around to yell an order at the baboons, the bristles are up with her hand near them.
When Oz flees the circus, he escapes by cutting the tether rope that the circus strong man is pulling towards himself. However, in the previous shot, there are clearly two ropes holding the balloon to the ground.
The map of Oz shown at 1:36 is backwards according to previously established maps of Oz. Purple Gillikin country and red Quadling country are in their correct locations of North and South respectively. However, yellow Winkie country is shown in the East while blue Munchkin country in shown in the West. Munchkin country is supposed to be in the East and Winkie country is supposed to be in the West. This is also established in previous depictions of Dorothy's Oz adventures as she liberates the Munchkins from the Wicked Witch of the East while the Winkies are liberated from the Wicked Witch of the West.
When Oz is telling Glinda his battle plan, he hands her a book. She thumbs through the book. In the process, you see the exact same page appear twice. It is the one with the trick box diagram. The wording on the pages is exactly the same.
The basket of the balloon is about waist high to Oz. It falls into the water and sinks completely. However, when Oz stands up the water only reaches his mid-thigh showing it is much too shallow for the basket to completely sink.
The machine which projects a giant image of Oz is based on a 'praxinoscope', and indeed the mirrors can be seen rotating behind him. However, that mechanism is totally irrelevant since what's required is one large stationary mirror in the light path between Oz and the lens to invert the image.