So basically it depends on how well you have carried out your SEO strategies to get top ranked in most of the search engines. Its easy to get a top rank on google and its equally difficult to be on bing or yahoo as all of them have totally different kind of algorithms.
google and bing run on different algorthims, so there's no correlation between the two, but i can say that all my sites rank higher in bing than in google, so if i focus on the google rankings and they are good, then i know that bing results will be even better! still not much traffic from them though, compared to google
There is something i notice on bing very big improvement. The locality search and keywords is precise. Unlike Google even it provide targeted search results there are still random result appear, but bing is different. For example if you search for classified ads it will be 90% all classified ads result will be listed starting from page 1 consecutively to the rest of the other pages. There is big chances bing can someday be Google closest competitor because some people looking for precision search. However it will take sometimes for users to get to know Bing, since they are too much believe that Google now is the best of the best.
Maybe yes maybe no... if it is visitted often, then maybe it is yes... but if the site just depend on the SEO, maybe it will different because they have different algorithms...
Generally back the front to what you have explained... if google holds your site high, bing usually follows (albeit months after)
I guess there is no connection between Google and bing, they are totally different from each other then how could Google will follow bing but yeah bing will definitely follow Google's index pages.