Whether you use "Süd", "Süden" or "südlich" depends on the context of what you are trying to say.
First, to directly answer your question:
Which is correct: "Ich komme aus dem Süd" or "Ich komme aus dem Süden"?
"Ich komme aus dem Süden" would be right.
And how exactly would you translate "Wir sind Süden"? "We are the South" or maybe "We are Southern/Southerners"?
This depends on the context. "Wir sind Süden" probably wouldn't be said in normal speech. If someone were to say "We are the South", they would probably say "Wir sind der Süden".
If they said "We are Southerners" there is even one more destinction:
If they were trying to say that they are from the south of germany, they would probably say "Wir sind Süddeutsche". Which uses the prefix "Süd" and combines it with "german". So basically, "We are south-germans".
If they are trying to say that they are from a southern country (e.g. Italy and Greece are considered "southern" in europe), then they would say "Wir sind Südländer", which means "We are south-landers".
Ill try to explain the basic rules behind this, although Ill have to say, take this with a grain of salt, as Im not 100% sure about the accuracy and the perfect grammatical reasoning. Just from the perspective of a native speaker:
Saying things like "Wir sind Süden" sounds a lot like a slogan. It feels like they try to say "We are the essence/heart of the south" or "We represent the south" or "This is what the south is like". The reason for that is, that one would usually use an article in that sentence. "Wir sind der Süden". This feels more like they are identifying with a specific, southern area.
Its like a company saying "We are best" instead of "We are the best".
"Süd" is usually not used alone. Nobody really says "Ich fahr süd". Cant say if that would 100% be wrong, but nobody says it like that. One either says "Ich fahre in den Süden" or "Ich fahre südlich". Here, the first one is usually used to say that your destination is some southern location. Like Bavaria in Germany for example. Whereas "Ich fahre südlich" is usually used if you are trying to describe your path, when "the south" is not necessarily your destination. Its basically saying "Im takint the southern road" or something like that. "Süd" itself is only ever used as a prefix or suffix.
- "Süddeutschland" = Southern Germany
- "ALDI-Süd" = The southern branch of the "ALDI"-company.
- "Süd-Stuttgart" or "Stuttgart-Süd" = The southern part of the city "Stuttgart"
I cant really tell if there is a "rule" to whether it has to be used as a pre- or suffix. As far as I can tell, thats pretty arbitrary and basically depends on what sounds better. In the examples given, it seems like locations or areas can use it with either prefix or suffix, like Stuttgart, and it wouldnt sound odd.
For "Names", like the name of the company "ALDI", it would sound odd to say "Süd-ALDI". Here only the suffix works.