We've got a Primary and Secondary Server (as a standby currently in Read Only mode ) on a Log Shipping set up. For the last week the TRN files have been backed up and moved across to the Secondary Server - but I've discovered they haven't been restoring.
In other words , I have a bunch of TRN files sitting in a folder and that need to be restored. 1)Is it possible to catch up , while the servers remain in their existing modes?
You need to check on the secondary server why the logs aren't restoring. More than likely, someone broke your log chain LSN sequence and it can't restore the logs after the break happened (that's usually the reason).
Log backup files may sit there for several days after restoring, depends on your settings in log shipping. You can look at log shipping monitor to find out if db is in sync.