Oscar Wilde's "The Importance of Being Earnest" is what is sometimes called a "drawing room comedy", set against the backdrop of the social interactions of the upper classes. Many elements of the play serve to signal the socioeconomic status of the characters.
First, one should note that the term "tea ceremony" refers to a uniquely Japanese practice which does not really have an English counterpart. Rather than being a "ceremony", afternoon tea is a meal, albeit one surrounded with social conventions. It is usually served in the afternoon, especially on days when the partakers would plan to have a late dinner. The foods being served represent the traditional sandwiches and sweets that constituted an upper class tea during this period, as did having a butler present to do the actual serving. This shows that Algernon despite his extreme modernity is actually in many ways a conventional member of the British upper classes.
The cigarette case is to a certain degree merely a plot device, but it also shows Jack to be a "modern" young man, as smoking cigarettes was considered very fashionable (and even daring) in the period. Its being made of engraved silver also suggests wealth.