Just think of the kinds of questions that philosopher ask--about goodness and justice, knowledge and belief, meaning and reference, just to take a few samples. There are no roadmaps for answering the kinds of question that (as Nicolas D. Smith said) set you to wondering. Even what counts as relevant to an answer is up for grabs. Questions recognizbly similar to "Does X exist?" "What is the nature of X?" (where X is usually something abstract and very general) seem to be asked generation after generation--but with different standards and criteria for what governs an acceptable answer. Our changing ways of doing philosophy, I think, partly account for the absence of definitive answers to philosophical questions. Our changing ways also partly account for the perennial interest in them.
Why do philosophers become Philosophers, is it purely intellectual or is it because all they are good at is thinking, and why for that matter aren't they out, thinking up the answers to the world's problems?
Although I agree with Alex about there being no general answers to your questions, I want to emphasize one point. Many people think that philosophical problems have no practical import or that it simply doesn't matter whether there are philosophers or not. Now it does seem that a society has to have reached a level of well-being in order for philosophy to flourish: if most people are starving or are living in fear of physical danger, they are not likely to produce people consumed by philosophical worries. However, it doesn't follow that philosophy is just another luxury to be regarded as dispensable. I honestly believe that philosophy is a major contributor to the value of a society. Think of Sparta and Athens. Sparta defeated Athens, but which society was more valuable? In my opinion, Athens (with its philosophy, sculpture, tragedies and so on) contributed more to civilization than Sparta (with all its military might). Solving "the world's problems" is not the only way to make a contribution.
Is philosophy like art? Is it a personal journey, where the philosopher finds a gnawing within themselves and seeks to unravel it using words and ideas? And the papers and articles they produce are artefacts of the journey - like stone markers they travel past on their way to somewhere?
Or is philosophy like engineering? The papers produced are like buildings, constructed using the materials of ideas and theories and the tools of logic and thought. The philosopher is more like an architect - working out what goes where and how it fits together to make something worthwhile.
I think that both the analogy to art and the analogy to engineering are good ones for philosophy. Different philosophers are motivated by different concerns. I would put Nietzsche on the "art side" and Kant on the "engineering side." Even so, Kant or anyone on the "engineering side" may be motivated in part by the gnawing that you mention. So, the two approaches need not be mutually exclusive.