Hi @bmarotta,
Allow me to join this thread and share insight on your query about banking in QuickBooks Desktop (QBDT).
Yes, you can delete those transactions found in the bank's register since they are duplicates. Moving forward, you can match your deposit bank feeds in case you already recorded payment for an invoice. Otherwise, you can add that bank transaction.
Additional details on these 2 processes are in this article: Add and match Bank Feed transactions in QuickBooks Desktop. It also includes a checklist on how you can review downloaded transactions.
When you're done, don't forget to reconcile your bank account to ensure its balance is the same as your actual bank account. See this article for details: Reconcile an account in QuickBooks Desktop.
Place a comment below if you need further assistance. I'll get back to you as soon as I can.