Hi,
One problem is that there's a CSS stylesheet linked with a relative URL, which doesn't work correctly with SEF URLs (not just JoomSEF, any SEF): modules/mod_hot_accordion/tmpl/hot_accordion.css
You can fix this by setting the "Set page base href value" option to "Yes - always use only base URL" in JoomSEF's Configuration on the Advanced tab.
Another problem is that your home page Itemid is being used for the search. You could probably fix this by creating a hidden menu item for the Search component in your Joomla. First create a new menu called for example "Hidden Menu" and then create a new menu item in it for the Search component. Leave it as published, but don't create a module for it. This menu item should then be used for search results, so you can assign any modules you want for that specific menu item.