First of all, yes, I think Bing rankings do matter, as well as Yahoo! and other search engines. As mighty as Google is, things that seem untouchable can change, as we have seen throughout the history of the Internet. Following the Bing Webmaster Guidelines (http://www.bing.com/webmaster/help/webmaster-guidelines-30fba23a) is likely to have a positive effect on your website, not just in Bing, but also in Google!
Second, I think there is a relationship between Google rankings and Bing rankings, although its probably not an intentional one - i.e. The Google Webmaster Guidelines (https://support.google.com/webmasters/answer/35769?hl=en) and the Bing Webmaster Guidelines have many technical, design and quality similarities, and they also share many algorithmic similarities.
However, a website can have high rankings in Bing, or Yahoo!, even rank number 1, and conversely not appear in the top 50 SERP entries on Google. This is where the 'quality' aspect of Google comes into play, for example where Google takes a much more serious view of unnatural incoming links, an subsequently the quality of websites listed in Bing is likely to be of a lower quality than that of Google. Equally, try signing up for a Google email account and you are frequently asked for your mobile phone number in order to confirm your identity. This is not the case on Hotmail (owned by Microsoft, like Bing). How many times have you heard of a hacked Google mail account? I haven't. How many times have you heard of a hacked Hotmail account? More than I have fingers and toes.
The key concern, in all of this, as a website owner, is traffic, or more specifically, converting traffic. A site can rank top of Bing and get low traffic, or 8th on Google and get massive traffic. In your analytics program, you will see your traffic, split between search engines, and you can see what percentage of the traffic for each keyword, for each search engine, has the highest percentage of converting users. Increasing traffic from whichever of these sources is most prominent will net you the most significant results.
