How does iCloud Photo Library's "Optimize iPhone Storage" setting work in relation to other photo apps on iOS? If I have Optimize iPhone Storage enabled on my iPhone and my phone storage is completely full, I understand that iOS will only store a low-res thumbnail of many of my photos on the device. (The full high-res version of the image will only exist in iCloud and on devices with Download and Keep Originals enabled.)

But when only thumbnails exist on the device, how does that work in relation to other photo apps on iOS? For example, do third-party cloud-based photo apps (like Google Photos, Dropbox's Carousel, or Amazon Prime Photos) have access to the full high-res images from the iCloud Photo Library? Or will they only see the low-res thumbnails in some cases?
It depends on where the app is pulling images from. If you have selected "Optimize iPhone storage" then these apps will only have access to the lower-resolution thumbnails, as bmike pointed out (they use Camera Roll). 

Unless the app is specifically coded to pull photos from iCloud Photo Libary (which Apple currently supports), they will just have access to the low-resolution photos stored locally on the device.

All the apps you have mentioned access your Camera Roll, therefore the low-resolution images.