May I use my non-electric coffee maker on Shabbat? I plan to pour "hot" water (from a Kly Shlishi, so there’s no problem with bishul-cooking) over ground coffee which sits in what is essentially a strainer, with or without a paper filter. Is my use of the coffee which drips through a violation of straining or the like?

Pouring hot water from a kli shlishi on coffee in a strainer is permitted because you are pouring in a liquid free of any visible particles and are hoping to get out the same thing- a liquid. What is forbidden is pouring a mixture of liquid and particles into a strainer to separate the particles from the liquid (Mishna Berura 319,33; Shmirat Shabbat K’Hilchata 3, 58).