Watson falters, an instant, as he opens the lid of the musical box after removing it from the biscuit jar, causing the lid to mostly fall closed again for a second, before he lifts if open fully. Yet, the tinkly music continues to play seamlessly, without any interruption from having the lid closed again momentarily.
It's extremely doubtful Holmes would be playing "Danny Boy" on his violin, a sentimental ballad more associated with the Irish community and popular with American film audiences at the time. It's not mentioned in any of the stories where he played classical music such as by Mendelssohn, Offenbach, and Paganini.
Dr. Watson refers to having met the King of Bohemia, which does not make historical sense, as the monarchy in Bohemia (land of the Czechs) was abolished in 1620. This is an allusion to the source story "A Scandal in Bohemia" which gave that land a tongue-in-cheek fictional history and politics.
The premise of the film is that the only way the prisoner in Dartmoor can communicate with his colleagues outside the prison is through the music boxes. So how does he get the information to them about when and where the boxes are to be sold?
When Sherlock goes to see the auctioneer to ascertain who purchased the music boxes, he reads the purchasers address from the log and says its on "Hampton Way" In the next scene he visits the house on Hampton Road. He also mentions the "correct" street, Hampton Road in a subsequent scene.
The auctioneer read the address from the log to Colonel Cavanaugh (the villain), not Mr Holmes. When Holmes and Watson visited the auctioneer, the dialogue had Holmes mentioning the (incorrect) address.