How can I show a slideshow of pictures in multiple subfolders while keeping them in order?
While iPhoto may work for some, it doesn't come bundled with every Mac and it's not free. It also relies on a library.

The quickest, cheapest (free) and easiest way to view/slideshow a folder and its subfolders is using Sequential. Just drag it to your Applications folder, run it once so it shows up on your Open with menu, right click on a folder, choose Open with > Sequential and hit the 5 key (for a 5 second slideshow).

Check its preferences if you want it to always start in full screen and its menus for aditional hotkeys (show thumbnails bar or skip to previous/next subfolder, for example). Besides supporting images, it also supports PDFs and archives (.zip, .cbz, .rar, .cbr) and you can choose to display them sequentially (folder's default order) or randomly.