Those are really great questions. I think quality content and how we are helping others and solving the answers to questions in our niche through our sites really takes the "authority" into play. Coupled with the more and more your site is on line also I believe helps to. I don't think there is no set number. Just continue focusing on helping others through your niche and the rest will come :)

I do have similar questions. I am not convince that it has to be a certain number of posts...yes we have to provide a lot of content, probably more than our competitors. I totally agree with ECV82. It will require us knowning, and sharing that information with our readers. Our blogs will be representing us. We will have to put the time in, and present in-depth information, explanations, and conclusions.
I believe that if readers find our writings to be informative and reliable, they will return to our blog, and they may even recommend us to others. This will help to improve your ranking.

Imagine you were looking for stock news on TV, you probably wouldn't go to CNN. I'm not saying you wouldn't get any stock news on CNN, if you wanted more in-depth information you would probably go to Bloomberg News

Those are excellent questions. I've frequently been in situations where I am talking to someone senior to me, but they are talking about the small process where I was the authority in the organization. I got there by reading everything I could (there were no training courses). Since this process applied to every employee, it was vital for the organization and management to understand. I would spend sometimes half my day explaining to individual supervisors how my process applied in their particular situation. I was the "go to" person and an expert in my field.

I would approach any niche like that. I would read up everything I could. If someone approached me with a question I could not answer, I would tell them that I needed to go research that, which I did and then gave them an authoritative answer. Many niches have more than just reading material available so I would avail myself of these materials in order to become an expert.

As far as the number of posts in my blog, I think it really revolves around how many real answers are you giving people when they need them.
Ellen