Hi,
I've checked your website and the cart refresh works fine.
Once I add a product in the cart, if I add a second product to it, it refreshes directly.
So the problem is not with the refresh system of the cart.
And if I don't have any product in the cart, instead of the standard "the cart is empty" message that should appear on the page, there is no message at all.
So the explanation of the issue is simple: As there is nothing to display in the cart module, your template doesn't display the module area at all. And thus, the refresh system of HikaShop cannot update the div of the cart module as it's not being displayed.
And this only happens because you have activated the "Use AJAX when possible for add to cart buttons" setting.
If you turn off that setting, the whole page will refresh when you click on the add to cart button and the cart module will be updated.
So either put back the empty message in the cart module, or turn back off that option and it will work properly.
And regarding the deletion of items from the cart, I actually don't see any delete button. I suppose that you turned off the "show delete button" setting of your cart module, or that you (or your template provider) customized the view of the cart module in a way that removes it.