If a dog is checking out early but the invoice was already paid, what happens to the two days that were paid for?

In this case you have to manually refund those if that's what they want.  We recommend creating invoices at checkout only so they're accurate. This is how most places do it.

If they select that it’s an early check out will it just change the days then and mark that run as now open?

If they've already paid the invoice, and they checkout early and click YES this is an early checkout, KC will adjust the dates on the appt.  but nothing happens at the invoice level because is already created and paid.

However, if they HAVEN'T created the invoice yet, and click YES this is an early checkout, it will update the appt and the invoice to the new number of days and charge accordingly.