He was so angry that he picked up his axe.
Is "that he picked up his axe" a noun clause?
He was so angry [that he picked up his axe].
The bracketed element is a declarative content clause functioning as complement of the adjective "angry".
Although the complement clause follows the adjective, "angry", it is actually licensed by the "so" that modifies "angry", and hence it may be called an 'indirect complement'.
I would strongly advise you to drop the term 'noun clause'. The classification of finite subordinate clauses is based on their internal form rather than spurious analogies with the parts of speech.
Yes. Another term for "noun clause" that you may use is "content clause", and this is a declarative content clause, headed by the conjunction "that". It is the complement of the predicative adjective "angry", and it means the result or effect of his anger.
"Content clause" is probably a better word than "noun clause" here, because while many content clauses can be thought of as functioning like nouns (hence the term "noun clause") complements of adjectives are not usually nouns or noun phrases but content clauses, or infinitive clauses or prepositional phrases. It does provide content to the independent clause, and it is not a relative clause.