Does moving an app to external (SD) storage also make it write application data and cache to external storage?
It does not make the app use external storage. The built-in move to sd functionality only moves the application package file itself to the external storage. 

If you want the app to not use the internal storage you'll need to root (and possibly install a custom rom.)
Depending on your rom you'll have several options to do what you want but virtually all of them require you to partition your sdcard in a special way (usually the custom recovery can handle this.)