Can a file be copied to one hard drive in a RAID 1 mirror set (software RAID), and later have the 2 drives sync up?
You're thinking of RAID-1 in the wrong way. RAID-1 is not an architecture intended to happily use a single surviving drive within a raid-1 group, and work out mirroring content later. Ultimately it will do that, but that is considered a recovery operation from a failed state, and it will destructively overwrite the mirror drive after replacement. In RAID-1, the two drives are expected to be both online and available all the time, as transactions will be committed to both. A loss of one drive will not stop the system from operating, but that is considered a failure mode, and as you've seen it will operate in a degraded state. 

The expected user operation at that point is to replace the failed drive, at which point the RAID software will make a block-for-block copy from the surviving drive onto the new drive, a process sometimes referred to as resilvering. It doesn't look at files within the filesystem. This can be a resource intensive process, and as you note, can take hours. This is expected behavior.

If you want something that will enforce eventual consistency in drive contents, use something like Rsync. There's an app to help set it up. http://arrsync.sourceforge.net