I'll answer the ones I know:
The three Save (stats, player stats, economy details) are for the stats files that get saved along with your game.
InGameServerMigration is if the host in an MP game quits, the game attempts to migrade to another host. If it fails, the game ends, if it succeeds another player becomes the new host and the game continues.
MP max pause count is how many times you can pause an MP game.
Background texture loading speeds up loading times because it loads textures as it needs them, not all at once (I'm pretty sure this is right, it was explained a long long time ago so memory is a little fuzzy).
WatchedFilesOn means the game will scan for changes in the entity files and try to apply the changes immediately. Mostly for modders messing with various values for units, it allows them to modify stats/etc on the fly without having to quit the game and restart each time.
LogOutput and ShowErrors are mostly for mod debugging. LogOutput should create a log file with errors, and ShowErrors shows which line in which file caused the game to minidump - commonly used for troubleshooting when mods crash for the mod makers.
The ones I'm iffy on:
The checksum settings should be for multiplayer syncing.
EmpireWindowBackdrop I think is actually a setting within the game that you can toggle, I think it either makes it a solid black or transparent.
