Just moving sites to Shopify and we have PayPal Express enabled haven't had an order by that method yet. On the credit card side via the Shopify API I pull the order into NetSuite, and create a customer deposit in Netsuite for the payment. Fulfillment's get sent from NetSuite to Shopify via the API.

We don't use Shopify, but do use Magento and PayPal (as our primary CC processor as well). We bring SOs into NS and once fulfilled bill them as Cash Sales. Avoid the deposit step and since we fulfill so quickly typically not a big timing issue for us