Why CAN'T I change the settings in Flash Player Setting Manager?

Chrome is it's own beast, so settings aren't going to propagate to/from the Flash installation in Chrome.  This is a design restriction imposed by Chrome, and stems from the fact that Flash Player only has mediated access to the filesystem.  Flash thinks it's reading and writing to the shared settings file, but it's actually being redirected transparently to a Chrome-controlled location on the filesystem.  You can set Chrome-specific settings by right-clicking on some Flash content and choosing Global Settings from within Chrome.  If Chrome's settings are also reverting after explicitly being set, you have two problems to solve.

The other browsers *do* use the shared settings file, and changes should be respected.  The vast majority of the time when I see these symptoms, it's a self-inflicted problem.  It's typically the result of a third-party "privacy" product that deletes "Flash Cookies", but they also nail the global settings preference file (settings.sol) in the process.  Since our preference file with all of your settings has been deleted, all we can do is create a new default one at next launch.

If you're 100% sure that this isn't the scenario you're in, then most likely the filesystem is corrupt, or we're otherwise unable to read or write to that file (again, outside of filesystem corruption, it would boil down to some third-party interference with normal operations).

If you think the file is corrupted, the first thing you should do is backup any critical data on the machine.  Filesystem corruption can be an early indicator of disk failure, and the next step is going to give the disk a good workout.  Better safe than sorry.  There are plenty of free/inexpensive online backup services, plus the traditional offline backup routes like external disks, etc.  I highly recommend this step.

Once your backup house is in order, you'll want to scan for and repair any problems with the filesystem.

Check your hard disk for errors - Windows 7:

https://support.microsoft.com/en-us/kb/2641432

At that point, you should be able to just delete the file.  Deleting and recreating it probably isn't going to help if there's an underlying filesystem problem, and is effectively what would have happened already if you've run the uninstaller and then reinstalled Flash Player.

See if you can delete the following file:

95/98/ME/2K/XP: C:\Documents and Settings\username\Application Data\Macromedia\Flash Player\settings.sol
Vista: C:\Users\username\AppData\Roaming\Macromedia\Flash Player\settings.sol

If you can't, the filesystem is definitely messed up.  I usually have good luck with Unlocker, but there are a few utlities described here:

http://www.guidingtech.com/10175/tools-to-delete-locked-files-in-windows/