the example: The people "responsible" for these feats were called Sumerians.
Is "responsible" used correctly in this context?
Are there any exceptions to the rule that would permit the use of adjectives after nouns?
The people "responsible" for these feats were called Sumerians.
Yes, it's fine. The adjective "responsible" modifies "people".
By no means all adjectives can be used postpostively. Clearly "responsible" can, as can "useful" (everything useful), “elect” (the president elect), "rich" (somebody rich), and a few others.
But the vast majority of adjectives are restricted to either attributive or predicative use, or both, so you need to be aware of that. You may be able to Google up a list of adjectives that can be used postpostively.
The people responsible for these feats were called Sumerians.
This is not an adjective used to modify a noun.
It is an adjective used in a clause:
The people [who were] responsible for these feats were called Sumerians.
This is a restrictive clause: the entire phrase "[who were] responsible for these feats" is adjectival and describes the people.
To say The people responsible for these feats... is a short way of saying The people who were/had been responsible for these feats. We use this shortcut all the time in English because people understand that the verb has been elided.
However, while it's the normal practice in English to place adjectives before nouns, it's not an iron rule.
It's common, especially in poetic or theatrical texts and with double adjectives, to say things like: A man tall and strong entered the castle at dusk or A sword sharp and gleaming was handed to him.