In Italian, if I am referring to Southern Europe, I could say Sud Europa (literally "South Europe"), or Europa meridionale.
Is South Europe acceptable in English, or could I say "the south of Europe"?
In Italian, if I am referring to Southern Europe, I could say Sud Europa (literally "South Europe"), or Europa meridionale.
Is South Europe acceptable in English, or could I say "the south of Europe"?
Southern Europe is generally the preferred term. The south of Europe and South Europe are sometimes used.
See Google Ngram Viewer: Southern Europe, South Europe, the south of Europe, the South of Europe:

As far as other names are concerned, the only rule is actual usage. Usually southern is used when referring to a region and South is part of proper names, but not always.
“North” or “Northern” as adjective
While in your specific case it's "Southern Europe" and using "South Europe" would be quite ...irregular, in general the rule is that there is no rule. Each place can be named North something, or Northern something, or something North, and it's completely up to the person who named it like that to choose.
Of course once given name is established, it's pretty much fixed from then on; you can't freely switch between North Korea and Northern Korea.