Where should I save images to have them appear in the Gallery app? Running KitKat, I want to extract a zip file with photos and place them in a folder where the Gallery app will find them. I'm using Android File Manager and Easy Unrar Lite to move files around and extract them. Where in the file systems should I put them?
It doesn't matter where you put them. A convenient place usually is the (internal or external) SD card, where you simply could create a Photos folder, and extract your photos to.

The gallery app simply checks the devices media database, it doesn't check the directories directly. This database gets fed by the so-called Media Scanner, which gets triggered by different system events, such as immediately after a completed boot, or whenever the SD card gets mounted. This is why you don't see the extracted images immediately. See my answer here for how to trigger it "on demand" (hint: for Kitkat, try the SD Scanner app).