We need to pay Sales Commissions based on Sales Invoices paid date. How to run a Customer Sales report that will tell me exactly what invoices were paid in a specific date? 

To do this job, you can run “Sales by Customer Summary” report.
You will need to customize the report to show daily sales and change to cash basis.
If you are using custom fields for sales rep on the Invoice, you can match the sales by Sales Rep as well.
See attached screenshot photo1, photo2 and photo3 for reference.