George Zucco confirms that while it didn't kill him (being set afire at the end of The Mummy's Hand), it "maimed and distorted" the mummy, which neatly explains why Lon Chaney looks nothing like Tom Tyler, it does not explain why the wrappings on Chaney's mummy are not even singed.
Partway through the movie (at around 34 mins), John Banning refers to dust marks on the throats of Aunt Jane and of Jim (the dog-keeper). But Jim was not strangled by the Mummy; he fainted, and the Mummy contemptuously walked by him, accidentally kicking earth against his face. So there was no contact between the Mummy's wrappings and Jim's throat. Just possibly there could be some dust on Jim's forehead, as it looks as if the Mummy's foot glances off it as the Mummy walks by. But certainly there would none on his throat.
When the Mummy is first awakened and travels to the Banning house, one of the neighbors is awakened by a shadow moving across her face. She wakes up her husband to tell him. He then looks out the wrong window to try spot the cause of the shadow. The moonlight comes in from a different set of windows.
Near the beginning of the film, Stephen Banning recounts his encounter with the mummy to his family. While Banning is narrating, clips from the previous film, The Mummy's Hand, are shown in the form of flashbacks. Banning is able to recount the final conversation between Professor Andoheb and Dr. Petrie despite the facts that nobody else was in the room with them, Dr. Petrie was killed by the mummy before he could leave the room, and Professor Andoheb was wounded and presumed dead. Even if he was able to piece together what happened to Dr. Petrie based on the evidence, Banning would not have had the perspective to recount the conversation.