My customer placed an order on my Shopify store and paid for it. Will this order be automatically paid on the BSDropshipping app as well?
No, it won’t. Shopify and the BSDropshipping app are two different, independent systems. Orders paid for on your Shopify store are not automatically paid on the BSDropshipping app. If you want BSDropshipping to fulfill the order, you need to go to the BSDropshipping app and pay for the order manually.