Cart image

Yes. You can add some CSS to display an image in the cart. You can use the CSS class hikashop_cart_module...

Something like this (You will have to adapt the CSS a bit to your case...add margins and/or padding):
.hikashop_cart_module{
background-image:url('http://my_website.com/my_image.png');
background-repeat:no-repeat;
background-position:right top;
}

I gave you the code to add the image to the cart module.

If you want to not display the whole products details in the cart, you can edit the hikashop options of your hikashop cart module and turn on the mini cart option to have something like godaddy.

Go in the joomla module manager and edit the options of the hikashop cart module. There, you will have a link to the "hikashop options". Click on it and you will have that option.

You're probably using an old version of HikaShop.
Please update and that option will be there.