You're female right? Going on that assumption: It depends on what you mean by important questions, or more specifically it depends on what your goal is in asking such questions. If your goal is to find out if he's dangerous in some way, then yes, ask. If the goal is to find out what kind of guy he is in general, then no. In the latter situation, take into consideration that when put under the gun especially, NTs (assuming again) are more likely to answer a question with what they think you want to hear - especially in the world of dating.
You'll get a much better idea of who you're dealing with if you just go on the date and let him make the content of his character apparent through his actions and the things he says when he's not been put on the spot. If he's anything close to normal, he'll be trying to impress you with his words on the date - take a chance at disappointment and let him display what his idea is of what will impress. It could end in disappointment, sure, but it could also turn out to be something wonderful. It's a numbers game, it really is.
I'm not trying to say that someone I don't even know is necessarily dishonest or deceitful, but people lie like freaking crazy for all sorts of reasons that aren't necessarily treacherous in their nature or intent, and when you ask someone a direct question as us-all-people prefer to do, you're giving them an opportunity to lie to you. Giving a guy a quiz to pass to possibly screen him out ahead of time almost guarantees you'll get correct answers, not honest answers. Trust me, I know, I'm a guy, we're aware of this quiz. All of us.
Just in case you're male here's an answer: No. Absolutely freaking not, whatever you do. Break your phone if that's what it'll take to resist the temptation. Seriously.
Well, I do hope that helped in some way, shape, or form.