Is either an insurance binder or an application for insurance acceptable proof of flood insurance coverage if the proof of payment for the premium is also obtained?

It should be. If questioned, note that the coverage is in force from the date of application/payment (subject to the 30 day rule, if applicable) and that it sometimes takes a while for the hardcopy policy to be mailed.

To close the loan, you want to have a copy of the declarations page (showing the zone, $ amount, effective date) or you can get a copy of the application. Also, always get a copy of the check. You cannot get a binder for FEMA insurance.

In the new Guidelines book (not yet released - I have a draft copy) it states "Acceptable proof of coverage may be a copy of the Flood Insurance Application and premium payment, or a copy of the declarations page. The NFIP does not recognize an oral binder or contract of insurance"