as for page rank is concerned for non competitive searches I have seen pages from a new website rank in the top ten, for more generic and competitive terms once that page or home page of the new website received page rank then it started to rank well. correct me if I am wrong but I still believe that page rank still plays an important role in rankings whether it be a competitive or non competitive term. on the other hand you also have the grace period where google give a new site a temporary boost in rankings for a short period then your rankings might start to decline and as the site ages and you gain more back links then you slowly start to see a rise in rankings for most of your keyword phrases in general.
for how quickly links affect ranking I would assume once google finds the link to your site it may start right away but since there are so many things that might affect rankings you might not know if a specific link(s) to your site were the cause for a rise in ranking. I might be a minor algorithm change ect... you just cant know 100% for sure if that was it.