The store is modified and I've emailed through the files that are not standard. To fix this, you'd need to examine those and figure out what they're doing. It's doing some sort of cookie handling that is most likely interfering with the cart cookie in IE.
I don't think the js errors are the issue, but I've sent a screenshot of IE showing them in IE dev tools. The main one is caused by a js file included with the skin. You could examine it and see if anything leaps out at you.
The rest of the server looks in good order.
Rob