Has there been any quality research done on search engine optimization in terms of correlation between an optimized website and increased sales or company performance?
Not sure if this is specifically what you are asking but I will take a crack at it anyways, Basically, what I think you're asking is has anyone done a case study on the postive/negative results of SEO performed on a site, specifically in regards to conversions and the overall affect on business. Most companies who have had a successful SEO campaign are going to keep that information extremely private because they do not want the competitors to know any details regarding the work. I'm sure there is something floating around out there on the web.
As Brad Callen indicated, it may be difficult to get hard facts to buttress the correlation you want proven but i think it stands to reason that an optimized website will naturally have increased sales, profit and thereby improved company performance from the following logic. Optimized site will mean that the site will rank for a number of keyword phrases and if previously thoroughly researched, such keyword phrases will command relatively significant traffic volume. The more the keyword phrases it ranks for, the better optimized it is and the more volume of traffic it will generate to the website. It also stands to reason that the higher the level of traffic that pumps into a website the higher the sales (ceteris paribus - all things being equal), even though not in linear proportion. Other factors such as conversion rate which is largely a result of the efficiency the webmaster puts into his landing page/website now come in to boost or retard the resultant effect of traffic volume in generating the needed sales, profit and ultimately, improved company performance. Such information may however be available at online research centres but certainly not with internet marketers who would not be interested in "wasting their time" ascertaining and recording a more or less "certain fact".
Here's some anonymized research for you: a software company I've had minor involvement with was spending around $30K a month running Adwords, because they were not doing SEO. They started doing SEO, became ranked on the first page for 3 relevant, high-traffic keywords, and were able to stop running Adwords and maintain the same monthly sales volume. Your case study is: SEO saves this company $30K a month.
Organic traffic will always convert great. If you rank you can definitely pull your adwords spend back, but it is not always the case that you can completely eliminate it. You will see most large corporations a lot of times be ranked number 1, but also be in the first paid position. This is mostly due to branding purposes though.
Not sure about the studies between SEO and Website optimization. But for increased conversions you would want to search for Website/landing page optimization, Web copy, Conversion optimization... There have been lots of studies regarding heat tracking and so forth on this topic. Sorry I can't remember the specific websites at this time though...