"That is a very difficult question."
Mondai is a problem. It can be translated as 'question' with the understanding that it is something to be solved.
Shitsumon is a question. It can be as simple as 'what's your favorite color?'. Normally not too much 'solving' is required.
As far as tests go, usually mondai is used as it implies something you have to figure out (solve), whereas shitsumon simply implies you need to answer it.
There is room for ambiguity and the two do overlap sometimes. As long as you understand the root meanings, you should be fine. It is not so different from the English differentiation of 'question vs. problem'.
This = これ, This jacket = このジャケット.
When including the object to which you are referring you always want to change the これ・それ・あれ・どれ into この・その・あの・どの. This is a very important thing to remember.
So, その質問 would be correct. それ質問 would be incorrect. As there is no real equivalent in English that I can think of in this case ('this is good' and 'this question is good', both use the same adjectival form of 'this'), I suggest not overthinking it and just remembering it.
Some particles hold onto old forms of phonetic writings. を is not お, even if it is pronounced that way, へ is not え, and は is not わ.
それ質問わとても難しいです ⇒ その質問はとても難しいです
それは大変難しい質問です。Looks fine to me.
I have a feeling that they might not like it because 大変 and 難しい can be quite similar in the meaning of 'difficult' and it lacks the formality that DL is probably trying to instill.
That being said, there are examples of this combination being used: https://eow.alc.co.jp/search?q=大変難しい for example.