OE 8.1D has a new feature that allows to autoassign check number for Debit and Credit Card payments. It consists of a box that is checked every time payment information is entered.

Is it possible to put that condition on the Payment Method table instead of having to check the box every time?

Sorry, NO.

There has to be some user confirmation that payment was actually received, or the invoice will just get "lost", since it is the presence of a document number (or the box being ticked) that tells OrderEntry to process the cash.