From my point of view, you should ask as many clarifying questions as you need in order to feel you fully understand the client's situation. Any question which can impact the way you think about the problem, and therefore structure it, is important. Examples:
- "What exactly is product X?" - if you don't fully understand what a product/service the company/industry is selling is, you have to ask. it will actually look bad on you if you get halfway through the structure and it becomes clear you don't really understand what we are selling.
- "How exactly does this industry work?" - If you are unfamiliar with an industry and aren't sure how it works, ask about it - e.g. repair parts for cars - does this mean you ship cars to manufacturers, sell directly to consumers, etc.
- "What specifically is the client's objective?" If the objective isn't super clear (e.g. want to return to profitability) clarify this as much as possible. Even if the overall objective appears clear, further clarifying it (e.g. does the client have a specific timeframe in mind.
These are definitely not exhaustive, and every case will have it's own nuances and ambiguities.
As for when to ask, you should firstly clarify the prompt (essentially repeat it back to the interview to make sure you haven't misunderstood anything), then ask as many clarifying questions as you need, and then take a minute to lay out your structure.