The image map isn't working because you missed the hash from the usemap attaribute.
<img src="http://www.thomasjfrank.com/icons.png" alt="Social network links" usemap="[B]#socialNetworks[/B]" style="border-style: none;" />
I'm not sure why you used a map for that as those icons could have been in a list in normal anchors anyway
You'll have to be more explicit with your other problems and identify exactly which elements are mis-positioned because on first look I couldn't see immediately what was different between browsers. It's difficult when looking at a page for the first time to spot if something is awry unless its blatantly obvious