I think you can't install anything on iOS without logging in, and the app store is the only way to get software (without rooting). I guess you could use the built in apps though, and the camera...
The iCloud service is separate from the App Store service, and you can be logged into one but not the other, however you log in to both using the same AppleID. You can actually be logged into iCloud using one AppleID and the App Store with a different one, at least a family member had their phone set up that way for a while, for reasons.
If you set up a Mac without an AppleID you'll constantly get nagged every time anything that requires it tries to do anything, which is frequently.
