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From a search point of view I'm not sure if this would make a difference or not, although I can think of arguments for both.
We currently point to www.ourdomain.com/index.html, but I think I will try changing this to www.ourdomain.com to see if it makes any difference.
As for the extension used, I would be sceptical as to whether this is true nowadays. I know there used to be issues with dynamic pages etc, but so many websites are built in Coldfuision, PHP, ASP etc these days that to penalise you for not having a .html extension would be a little unfair.
Hope this helps.
As for the links to the homepage, remember that from the SE's viewpoints, example.com/ and example.com/index.php are different URL's, so if you have links to different 'versions' of your homepage, you are making the SE's pick a best or "canonical" version and you are also splitting the juice flowing to the homepage across different URL's.
Pick the one most likely to be linked to from the outside ... usually "http://www.example.com" or "http://www.example.com/" (even the "/'s" are technically different) and point all links to that one version. Then 301 redirect all other versions to the chosen URL, and you're set. :)