I've heard bad things about all my internal links that point to the home page (index.php) should be pointing to www.domain.com/ instead. I've also heard that having the extension php instead of html makes a difference.
In terms of pointing pages to index.php rather than www.yourdomain.com, I would suggest the main reason to possibly avoid this would be the work involved in changing links if you every changed the name of the homepage ... although most editors do have a find and replace facility these days.
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.
I dunno where this is coming from lately but it doesn't matter what extension your filenames are. .php, .htm, .html, .asp, .whatever. It's all fine.
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. :)