particle か - Indirect questions nested inside phrases

Trying to use interrogative words in nested (indirect ?) questions has been troublesome for me.

First came things like:

先生が何を言ったか知らない

Which I guess means "I don´t know what the teacher said". So far, so good.

However, I have been looking into using と for quotations, or for reporting to a whole phrase.

For example:

彼はこれは何だと言ったか

Which I think means "What did he say this is ?"

But what if I just wanted to know if he said this is something or not, like: "Did he say this is something, or didn´t he say it ?"

Would it be something like this ? (below)

彼は「これは何か」と言ったかどうか?

Lastly, if I wanted to say "Do you remember what he said this is ?", would it be:

彼はこれは何だと言ったか覚えていますか?

If there is anything wrong with my guessed meanings, please let me know...

Answer

彼はこれは何だと言ったか is not a complete sentence but a clause that's embedded in a sentence like 彼は（自分が）これは何だと言ったか覚えていなかった："He didn't remember what he said it was".

"Did he say this is something, or didn´t he say it ?" is expressed as あの人、これ が(/を) 何（か）だとか言ってた？ それとも言ってなかった？（or …言ってませんでしたか？） or so. Using 言っていた instead of 言った is another point. (This is difficult to explain, though)

彼は「これは何か」と言ったかどうか is not a complete sentence either, but a clause as in "He didn't remember if he said "what's this" or not".

彼はこれは何だと言ったか覚えていますか? more likely means "Does he remember what I/you/he said this is?" than "Do you remember what he said this is?". If you want to make it clear, the latter should be 彼が これ は(/が) 何だと言ったか覚えていますか.