The question is very broad, as you surely realize, since you broke it up into six sub-questions (which are also very broad!). In order to attempt an answer, I will have to send you on a bit of a scavenger hunt:
First, see my answer here and note that I have a literature review from my doctoral dissertation, which was entitled Conceptions of Creativity in Elementary School Mathematical Problem Posing. Practice with formulating your own problems is one way in which you can foster creativity; the literature review contains many references you can follow up on with regard to creativity, mathematical problem posing, or both. Moreover, you can find further remarks of mine about problem posing here.
Another potential way to cultivate creativity is to impose constraints on your work; to this end, check PD Stokes' (non-mathematical) book Creativity from Constraints and see if you find it helpful.
Other good sources are some of the work by B Sriraman and his collaborators; most recently, I received a recommendation for a book by P Borwein et al that I have yet to check out, Mathematicians on Creativity, which is described as follows:
This book aims to shine a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-caliber working mathematicians. In their own words, they discuss the art and practice of their work. This approach highlights creative components of the field, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at work.
If you like creative people's recollections, then a non-mathematical source (though it includes a nice piece by Feynman) is F Barron's Creators on Creating. Of course, one cannot always take self-reported stories at face value; a discussion of this matter and how storytelling relates to creativity can be found in work by MC Bateson.
More generally, if you are going to go with a single textbook to gain a broad sense about creativity and the different ways in which it is conceived of (especially as related to problem solving) see Weisberg's text mentioned in my answer here. Again, this is not explicitly about mathematics - in fact, there is a dearth of work on STEM creativity! - but you may well find tidibits of interest. (And there are at least some mentions of the hard sciences, e.g., genetics, biology.)
As far as whether people are inherently creative: No, I do not believe so, though any question about the construct will likely depend on the definition (or conception) you have in mind. One creativity theorist of whom I am particularly fond is Howard Gruber; he effectively defines creativity as purposeful work, and this is not something we are born able to churn out. Purpose is something that is developed over time, and Gruber works to "demystify creativity" as many others who use case studies are inclined towards. For a couple more of my answers that mention Gruber and his work, see here (less relevant) or here.
Though following the links above (and links therein!) will already entail quite a bit of reading, I thought I would voice one pet peeve of mine: Many studies in the mainstream media that are purportedly about creativity use the Torrance Tests of Creative Thinking (TTCT) as their main instrument. However, the TTCT really measures what is today known as divergent thinking (DT) and, for those who research creativity, this is no longer thought to be the same as creativity. In fact, such an identification should have been put to rest in the 1970s (!).
Even (or especially) major researchers of DT - such as Runco - will tell you this. For example, see (or at least glean the essence from the title) his Commentary: Divergent Thinking Is Not Synonymous With Creativity (pdf). One must be wary of studies that use TTCT-like assessments to measure DT, and then form strong (i.e., unfounded) conclusions about creativity.
If you are interested in other ways to assess creativity: I am somewhat fond of TM Amabile's Consensual Assessment Technique (CAT) for which you can find a strong overview article by Baer and McKool here (pdf) that has a fair bit in the context of higher education. Insofar as the above-mentioned TTCT is concerned, note that the authors write in their introduction:
If creativity is to be assessed in college settings in a meaningful way, divergent-thinking tests like the Torrance Tests of Creative Thinking and other commonly used creativity tests are inadequate because they fail to meet even the loosest standards of validity.
Not all of what is written in the mainstream media about creativity is as unpalatable as some of the TTCT based work. A recent piece by Brooks in The New York Times called The Creative Climate includes the following interesting snippet:
Today we live in a distinct sort of creative environment. People don’t so much live in the contradiction between competing worldviews. We live in a period of disillusion and distrust of institutions.
This has created two reactions. Some monads withdraw back into the purity of their own subcultures. But others push themselves into the rotting institutions they want to reinvent. If you are looking for people who are going to be creative in the current climate, I’d look for people who are disillusioned with politics even as they go into it; who are disenchanted with contemporary worship, even as they join the church; who are disgusted by finance even as they work in finance. These people believe in the goals of their systems but detest how they function. They contain the anxious contradictions between disillusionment and hope.
I find the above excerpt quite interesting for a few reasons. One, it relates to some of the other literature on creativity: the anxious confrontation alluded to in the last sentence reminds me of R May's work in existential psychology, and especially The Courage to Create (pdf). (The word monads also comes up in Darwin's work; cf. Gruber's book Darwin on Man.) Two, the notion of distrusting institutions is an important theme in Latour's An Inquiry into Modes of Existence, which is part of an on-going collaborative project of potential interest to those who study creativity. Its corresponding site, for which registration is free, contains the entire book; it also contains a video of a mathematician (darij grinberg) writing a question and uploading it to MathOverflow (the specific question can be seen here).
Perhaps it would be best to curtail myself here; my apologies for the lengthy nature of this response.