Hi,
1. Looking at the issue on your website, you must have the file templates/YOUR_TEMPLATE/html/com_hikashop/product/cart.php which is an override of the cart module. This override doesn't support the cart refresh system. Rename that file to deactivate the view override and it should work properly. Note that you might loose some styling on the cart module so you would have to redo a view override with the changes you had made before.
2. You can configure the checkout via the tab "checkout" of the HikaShop configuration. I would recommend to have several steps with the login/address views first and the payment / shipping in another step after, etc.