How to work out which cart is the users?

How can I quickly work out, with php, which cart is the users?

I'm not sure I understand what you want to do. Could you explain more ?

I was also thinking that if I could work out the cart_id I could create the user myself and load their details into hika shop directly and then just use hika shop to display this information.

Once a cart has been created, it's easy to get the id of the cart of the current user.
You can do like that:
$app =& JFactory::getApplication();
$cart_id = $app->getUserStateFromRequest( 'com_hikashop.cart_id', 'cart_id', 0, 'int' );

Create yourself the cart and the infos of the user in the database is not easy though as it involves at least 7 tables.