On HikaShop commercial versions there is an option to remove the name/username/password/password2 fields from this form. Just the email and the address information will be left so that the user has less fields to fill. Now, if you just want to remove the name field, Joomla doesn't allow you to register a user without a name, so what you could do is use the username field value as the name. For that, change the view itself with the display->views menu. You can search for "login_register". In the view code, you could remove the label, change the field type to "hidden" and add the code
onchange="document.getElementById('register_name').value=this.value;"
to the username field. That way the name of the joomla user will use the username field value. You could also use a concatenation of the other fields value but that requires a bit more javascript code.
You don't need the joomla login module. The login area you see on the HikaShop checkout do not use it at all.