I don't know if there's a "by the book" answer. Interviews, by their very nature are a bit scattershot and biased towards the... biases of the people asking the questions.
Personally, when I ask these sorts of questions, it's because I think you can't be critical or honest. I dislike canned answers. I dislike "I don't have any weaknesses" or "my greatest weakness is sometimes my awesomeness makes people uncomfortable" - everyone has weaknesses. In short, I want to hear you own something bad about you, or the fact that a bad coworker was just bad. I don't want to work with someone who is too scared to give me bad news, or ignorant of their own flaws, or who pushes their mistakes on to others.
Other people I've known want to see how well you tell a story. How can you take the bad coworker and tell a story that puts you in a good light - since you'll often need to tell a good story to customers/managers when things don't go smoothly.
Others still want to see what you consider a weakness or a bad coworker. If you spell out traits that mirror things they value as strengths... you'll probably not get the job.
In short - it depends. The fact that these questions are common doesn't mean they're any good. It means that many interviewers, when thrown into doing an interview think "oh, what common questions should I ask?". Since these questions can apply to everyone, they proliferate.