Do people ever fact check a public speaker while they are speaking just to see if they are truth tellers or not?

Answer 1

Yes.

Just watch cable news and you can see it happen in real time, though often the "facts" offered there are more in terms of (perhaps) informed opinion than empirical information.

But if you're presenting information, especially in front of a skeptical and/or hostile audience, you can count on the fact that they can — and will — use Google and Wikipedia in real time on their mobile devices to validate the information they may question in your presentation as quickly as you used it to research the subject.