What happens to the order if a customer closes the payment window or their online payment fails?

The order is still saved. Wcart creates it before sending the customer to the gateway, so it stays in Orders → Manage Orders with the status Pending. On checkout the customer sees: "Payment was not completed — your order is saved and awaiting payment."

The order only moves to Processing when the gateway confirms the payment. Their cart is not emptied if they cancel at the gateway, so they can try again.

For an unpaid order you don't want to keep, open it, set the status to Cancelled, choose Payment declined as the reason and click Update Status.