The Latin words that Holmes sings at the beginning are "Quicumque bonum desiderat caelum et fontes aquarum desiderat", which roughly translate as "Anyone who desires good, desires heaven and fountains of water."
The Latin music that Holmes is singing in the opening scene in Baker Street is the "Quemadmodum desiderat" by Orlandus Lassus. That's taken directly from Sir Arthur Conan Doyle's original version of the story, in which the polyphonic motets of Lassus were mentioned:
"One of the most remarkable characteristics of Sherlock Holmes was his power of throwing his brain out of action and switching all his thoughts on to lighter things whenever he had convinced himself that he could no longer work to advantage. I remember that during the whole of that memorable day he lost himself in a monograph which he had undertaken upon the Polyphonic Motets of Lassus."
"One of the most remarkable characteristics of Sherlock Holmes was his power of throwing his brain out of action and switching all his thoughts on to lighter things whenever he had convinced himself that he could no longer work to advantage. I remember that during the whole of that memorable day he lost himself in a monograph which he had undertaken upon the Polyphonic Motets of Lassus."
When Sherlock looks out of the window he says "It's a real pea-souper." This is referring to the pea soup fog, a fog caused by air pollution from factories that used soot and sulphur oxide. In the 19th century factories used an abundance of fossil fuels that contaminated the atmosphere with the thick fog of varying colours, which killed many from its toxicity. Changes were made in the 1950s to cut down on the use of fossil fuels after the Great Smog of '52.
Amanda Waring, who played Violet Westbury, is the daughter of Z Cars (1962) actor Derek Waring, who played Dr Watson in Sherlock Holmes - The Musical and Henri Fournaye in The Second Stain (1968).