Briefly, for now: the stories all have in common a skewering of the pretensions of the middle and upper classes, and all end with a twist (some more successful than others). To me, the main issue that separated the stories was whether or not the reader feels that justice has been served. The aunt in The Lumber Room,, for example--it's impossible not to take unholy glee in her predicament. But when one thinks of the potential repercussions in The Dreamer, it's highly unsettling even as you snigger.
In general, to me Saki reads like Waugh stripped of any pretensions of civility. Bile with no redemption. I enormously enjoyed him, perhaps even for that reason, most especially for the economy of his language. However, the tropes get a little tired taken en masse.