Can I set a slideshow background on iOS? I know that on a Windows PC I can set my desktop background to be a slideshow of images that changes periodically. Is this possible on an iPhone in any way? If so, how can I do it?
Short answer, no. iOS built-in feature set doesn't support a background slideshow.

App Store apps can't automatically change the wallpaper on a device, so you won't find a third-party app to do this for you.