Finally the downloading works.
As it turns out, the problem was in Android's standard Environment.getExternalStorageDirectory(); field.
It was returning an invalid directory, but I'm certain this is an issue only to my phone, or other SGSIII phones running AOSP.
Regardless, I will research more into this issue. However, I can use the path of "/sdcard/" in the mean time.
"/sdcard/" should actually work on every phone, but I would still prefer to have a sure working method on all phones.
I'm going to do a Spring cleaning of my code rather soon, it's getting a little too messy for my liking.