I do not understand the mysterious 24-hour rule. Does it mean for example: If I use my mixer on Thursday for making a dairy cake, can I use the same mixer for making parve mayonnaise on Friday, which I intend to serve with the meat on Shabbat dinner?

The 24-hour rule is complicated and there are differences to practice between Spharadim and Ashkenazim. Basically, the 24-hour rule states that anything absorbed in the utensil will be pogum, i.e. give a bad taste, prohibiting something prepared in it after 24-hours or less. This 24-hour rule is usually a bediavad rule and not to be relied on initially. Without going into details, if everything you use in your mixer is cold, i.e. you don't use anything hot or sharp in the mixer, then you can use the same mixer in the way suggested in your question. Otherwise, initially you shouldn't do so.