Our expert says:

CyberShrink

There is no right answer to such a question, and many potential wrong answers. The questions you quote would want to get an idea of how do you deal with conflict ( seek appropriate help, try to negotiate / or use nuclear weapons ? ) or How do you work out priorities when there are several potential priorities on your desk ?
As Purple wisely points out, they'd like to hear something based on an actual example you have encountered. If you have never met such a situation, say so, and ask whether they'd like you to try a more theoretical answer, or vary the question.
I remember when interviewing applicants to medical school in Canada, I asked one guy who said he had been working, in the previous 3 years, in a cement factory testing whether the concrete / cement mixtures they qwere producing, met minimum safety standards, what he would have done in a sitation where the tests showed the product to be unsafe, but his bosses urged him to fudge the results and declare the stuff to be safe for use in buildings. He looked a bit puzzled. He said, "But that's exactly what happened." What troubled me was not how he had chosen to handle the problem situation ( there could have ben a number of suitable ways ) but that he didn't even see that there was potentially any problem whatever - now THAT was indeed a wrong answer. If he couldn't even see a problem in a situation which could have lead to the death of innocent people, he was in no way suitable for us.