Learn French: Il demande ce que ce mot signifie.

In an indirect question, you use ce qui when "what" is the subject of the action, and ce que when "what" is the object. Here, the subject doing the action is ce mot (the word). Because "what" is the object (the thing being meant), we must use ce que.

Could I say Il demande qu'est-ce que ce mot signifie?

No, though it is a very common trap for English speakers! Qu'est-ce que is only used for direct questions (when you are asking a standalone question that ends with a question mark). When you embed the question inside a statement to make an indirect question, you must drop the est-ce and just use ce que.

Why does the sentence have ce twice in a row (ce que ce mot)?

They look identical but do completely different jobs here. The first ce is part of the set phrase ce que, which means "what" in an indirect question. The second ce is a demonstrative adjective meaning "this" or "that", pointing directly to the noun mot (word).

Is it possible to change the word order to ce que signifie ce mot?

Yes, absolutely! In relative clauses and indirect questions, it is very common and natural in French to invert the subject and verb if the subject is a noun (like ce mot). Both ce que ce mot signifie and ce que signifie ce mot are correct and widely used.