Dana:

in most cases you can tell what the current accounting period is via the the module options.  For example in general ledger-->setup-->general ledger options.  However, I have seen many people simply advance periods, so this is not a guarantee.  But at least you will know what the system thinks is the current period.  By default, period end processing would attempt to close that period when you run period-end processing, whether prior periods were closed or not.

Often information is also available in the System Activity Log, where a record will be written to show when period end processing is started or ended as well as the module. 

Also, if you use paperless office to produce period end reports, you can trace these reports to the correct folder for the particular module per your paperless office setup for period end reports and see the creation date of the last reports. 

I often look prior to closing periods to see for each module what the retention dates are for saving paid invoices, etc. to make sure that before closing a period, I will still have access to data that is needed for reporting.  Many people have not closed periods because they are dependent on some historical data that they do not know how to obtain otherwise.

You can copy the live company to a backup company.

History is purged during closing so review what your history retention is in each module before you close them.