Hi,
1. When you press enter on a page, the first HTML form of the page is submitted. It just happens that on your page the first form isn't the cart module. Some custom javascript could be added to make sure that it capture the enter key pressing when the focus is on a quantity field of the cart module, but it's not that easy to add.
2. The product code is displayed because you have the "display product code" option activated in the HikaShop configuration. Turn it off and you won't get it in the cart.
3. The cart module has a refresh icon next to the quantity field:
take.ms/FPSYMQ
So if you don't see it, it's either your template which customizes the display of the cart module and removes it, or the layout/CSS of the page which make it so that it doesn't appear. Without a link to the page where we can see the cart, it will be hard to say what the problem is and give a solution.