I've read multiple people having this problem but couldn't find a discussion related to a MacBook Air. I have the most recent version of Safari. Flash Player runs fine until I restart my computer and then it's completely uninstalled.
Yeah, it's very unlikely that Flash Player is uninstalling. Safari's preference file gets messed up from time to time, and it's probably a case of the preferences not being written to disk. They persist in memory while the application is running, but at the point that they need to be restored from disk, you're getting whatever the last successful write was.
First things first, I'd sanity check the disk and repair any problems discovered.
See the "Try Disk Utility" section:
While you're there, choose Repair Filesystem Permissions (Verify will frequently return clean, but then Repair will still fix a bunch of problems).
At that point, you might just uninstall and reinstall, and see if that's enough to get things working through a reboot.
Uninstall Flash Player - Mac:
Download Flash Player:
If your settings aren't sticking, you'll probably need to enable Flash Player again.
If that doesn't help, resetting Safari to factory settings will probably get the job done. This is destructive, so you're going to lose your history, cookies, etc, which is why I saved it for last. Once you do that, you'll probably also have to go through the motions of uninstalling and reinstalling Flash Player, and then enabling it again.
I can't find an official Apple guide on the topic, but this fits with what I was thinking about:
If you're still stuck, please let us know.