One can write a sentence with a partial quotation:
Direct speech: We live in a madhouse! We have to move.
She says they "live in a madhouse" most of the time.
This is unobtrusive: the quotation is direct speech, because the exact words are repeated, made to fit into the syntax of the sentence, which was easy enough. It is a predicate, so it fits ...
(1) I agree that the sentence you mark as wrong is much better without the that; (2) do not look to Wikipedia for good English. I would leave the that out in both sentences, the one you marked wrong and the one you quote from Wikipedia.
When reporting a speaker's words about the present or the future, we can choose to backshift into the past or to keep the original speaker's tenses. For example:
I told my boss I will/would be late to work tomorrow.
He asked me how old I was/am.
But as Swan in Practical English Usage (p277) notes:
We are more likely to change the original speaker's ...
The meta-humor lies in making reported speech into quoted speech, as a play on words. The original "That's what"—She renders as
"That's what," she said
Your alternative "That" — She reads as
"That," she said
She said, "that."
If you turn this around and make it indirect speech again ("That" is what she said), then it wouldn't be ...
What's the dinner? (not a very idiomatic question)
"the dinner" is the subject and "what" is the subject complement. Proof of this is that the answer will be something like:
The dinner is fish. ("fish" is the answer to "what")
What's for dinner?
"what" is the subject. This becomes evident in the answer:
Fish is for dinner.
In the embedded or ...
Both the sentences are correct. It depends on what context and which tense you are using it. For example:
Aristotle was the first person who described the shape of the Earth.
He said the Earth was round but no one believed him then. However,
today it's a known fact.
I asked my professor, "What is the shape of the ...