Great topic, I think it's a smart idea to ask questions, especially if you're concerned about risk, but in my opinion, with a change in approach you can get a more effective answers.
Most of your questions should be directed at the CEO, or at the very least a co-founder. Depending on the size of the company, the people you're interviewing may not know about funding issues and exit strategies.
Asking about Runway is a good question. If you're worried about coming into work one day and finding the office closed, ask point blank about your concern. This is a good test question to see if the company believes in transparency.
You're right, the company may not want to go into too much details on it's traction and funding, but they should be comfortable talking openly about your concerns.
As to new funding, you should be able to find out their last funding round, either through Crunchbase, or from SEC Filings. Startups tend to fundraise in the late spring, and the late fall. Come thanksgiving, investors are hunkered down for the holidays, and summer vacations make it difficult to get people together. So if you're approaching a company during those times, you can assume their prepping for a funding push later.
Despite what others may say, the CEO is always fundraising. Perhaps not actively, but working towards their next round.
Ask about milestones towards their next raise. If their a consumer app, what are their MAU targets. If their SaaS business, what are their Churn and MRR numbers. What are some other milestones that they're pushing towards?
Questions about potential exits aren't that interesting, because it's easy to judge a business that is poised for IPO, and those that are ripe for acquisition.
If the business has taken institutional money, then there really is only two viable outcomes: Acquisition, or going Public. Investors won't invest in lifestyle businesses.
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Your family's financial well-being hopefully isn't tied to a stint at a startup. Perhaps I'm a bit old school, but I believe in some amount of financial planning--something like a rainy day fund that can keep you afloat for 3 months without income.
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It's not just startups that are at risk for closing down. Irrational Games just closed shop and let go of all but 15 people, and that was despite releasing several AAAA blockbuster titles.
Losing your job at a startup isn't nearly as bad as you might think. There is a major shortfall in technical talent in most startup hubs, investors and founders are always looking for talent. After Zynga shut it's seattle offices down, a local VC here had an open house for the employees that were let go. So, if things go bad, many founders and investors will work to help secure you a position elsewhere. Plus, again, it's hard to hire people.