Academic job search: writing sample
1) is it appropriate;


I'd consider this the second best. Postdoctoral and faculty are mostly valued by how well and efficiently they can write and publish academic papers so having some published work to show is the best, and if there isn't any, then having an excerpt from the dissertation would also help.

However, do not just "extract" without making any change. Introduction in thesis may not be the nicest thing to keep readers' attention. Instead, use this chance to draft a paper, and write it out like a synopsis or executive summary. This way, you can showcase what you did in your previous study and also give the committee a writing example.


  2) are there any reasons to prefer literature review to introduction
  or vice versa;


The ultimate goals are to show your mastery of English and expressions of concepts and arguments. I personally do not think an introduction without any review is convincing enough, nor a literature review without any big picture is descriptive enough. Thinking as a committee member, I want to see a self-contained story rather than parts that are abruptly chopped off from bigger chunk of works.


  3) is there a benefit of doing so, even when a writing sample is not
  required?


There are two approaches, both I'd consider acceptable.


Acknowledge that you currently are working on publishing your works (so no publication yet,) then proceed to present a writing example for their reference.
Include it in the appendix, or in the letter explain that writing examples are available upon request.


The way I evaluate it is: does the whole application look tailored to what the committee asks for? You need to show that you have put together a package according to their ad, and not merely changing the school, department, and the chair's name in a mass production.


  I certainly plan to convert my dissertation into several papers
  and/or, perhaps, a book chapter, but it will take quite some time to
  do that.


Do it now. Really. Everything about writing will "take some time" and it's the famous last word for many researchers. Make the time and get them done. Once you're in the new position all those thoughts and details will be forgotten very quickly; the interest to publish them will fade and guilt will grow. Use this application time as a motivation. Postdocs or new faculty members with publication history does distinct themselves from the rest of the pool; please use this chance to make a start.