This translation is the most generic one. In pretty much all cases where enemy is appropriate in English, Feind would be in German. Most importantly, this includes the rival of your current warfare.
Wir werden dem Feind morgen bei Königgrätz begegnen, macht euch bereit für die Schlacht!
Two of the other words you found are direct modifications of Feind which only warrant a sub-entry in this post.
While not being able to back it up with sources, I think that the prefix Erz- originated from the word Erzengel (arch angel). I’m not sure about the development of Erzengel but it likely didn’t have anything to do with Erz but originated from a similar sounding word of a different language. And like the Erzbischof (archbishop), it was extended to other words being very important.
Thus, the Erzfeind is a very important enemy; one that has been an enemy for at least a generation, where various wars have been fought and where reconciliation seems very unlikely. Traditionally in German German usage, this applied to France (and pretty much only France). It is to be seen in the context of Bismarck’s anti-French alliances and the German-French war of 1870/1 plus the two World Wars (which each had a notable anti-French component).
If you wanted, you could translate it as arch enemy into English. I would be cautious about using it in German.
This is an enemy that you hate so much, you would fight him to either of your deaths. The compound part Tod-, meaning death, signifies that. Like mortal enemy, you can consider it a stronger form of Feind.
X ist nur mein Feind. Aber Y mag ich so wenig, ich würde ihn Todfeind nennen!
This is the root that forms most of the meaning of Erzrivale, which was included in your list. Its meaning is well translated as rival into English: A person who crosses your path, with whom you have to measure your powers but not more. The key difference between Feinden and Rivalen is that the latter might sit down for a beer after a winner has been determined, while a losing Feind would use such a beer to stick a dagger into his opponent’s intestines.
Thus, Rivale is often found in sports contexts. But for example in manga translations, a Rivale can also be a love rival.
This one basically follows the same idea as Erzfeind only without the historic connotations Erzfeind has. It is a rival you keep measuring yourself against (but against whom you don’t have to hold serious grudges).
Der Erzrivale des FC Schalke 04 ist Borussia Dortmund.