Went on your site again.
I added a product in the cart and now I have the product and the Flat rate is showing accurately.
I went back on you shop page. I added another product (different) and the Flat rate is still showing accurately...
I add a third one, and everything still ok..
In the cart, I added one quantity to the first product (for a total of 2) and updated the cart and everything is still ok!!!
So your cart is working well!!!
On chechout page, I have no message under Your order - Shipping. The only message I have is this one (because I haven't entered any billing address):
Sorry, it seems that there are no available payment methods for your state. Please contact us if you require assistance or wish to make alternate arrangements.
BUT...
I emptied the cart, and try to register to test the cart once registered.. but I can't... even if I enter a password an error message is appearing telling me to enter a password.