Can "that" introduce an adverbial clause of result？ Do the following sentences sound natural to you?
1a: What have we done that you should be so angry with us?
1b: “Are you starving that you must steal?” said John to his son.
Thank you very much!
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Clause of result is an ambiguous term: it appears that in teaching English it is usually used of so that and so/such...that clauses, which may describe either a result or a desired result—a purpose. In these the that is often omitted.
It was so hot (that) he took off his coat.
He took off his coat so (that) he would be cooler.
Bare that was indeed used to express purpose in older English, with a following 'subjunctive' clause:
He died that we might live.
She works that her children may not starve.
Today, however, we use so that or bare so, and the subjunctive quality is usually weakened or eliminated:
He died so (that) we would be able to live.
She works so (that) her children won't starve.
This use, with bare that, is now purely literary, and largely confined to religious contexts, where the influence of the 17th century Authorized Version of the Bible is still strong.
The two sentences you give are a little different, though they are grounded in the same idiom. In both, the that clause expresses a result, but it is not a desired result: it is an existing state for which the speaker seeks a cause.
What have we done that you should be so angry with us?
= Why you are so angry? What did we do to cause that?
Are you starving that you must steal?
Why must you steal? Is it because you are starving?
The first is fairly colloquial, though today a for clause would be a little more likely: What have we done for you to be so angry with us?. The second, however, is decidedly literary; in colloquial speech must would become have to and the question suggesting the cause would be recast with an explicit ‘cause’ term:
Is it because you are starving that you have to steal?
Are you starving? Is that why you have to steal?
Both your answers are correct, albeit extremely literary and outdated (almost Shakespearian).
More contemporary usage would be:
What have we done to make you so angry with us? -or- What have we done for you to be so angry with us?
Are you starving to the point that you have to steal? -or- Are you so hungry that you have to steal?