Using Quit and Save is known to cause issues with the game. Have you tried just using save or Save As?
If the mailbox isn't working on your current home lot, try moving the family to a different lot to test the mailbox. It may be that the lot has glitched some how.
Trying to delete and replace the mailbox messes up the lot and is not fixable as far as I know. When I somehow did that I didn't save and went back to a previous save. The other thing you can do is re-build the house on a different lot.
If you feel your save has corrupted you can save your household via Edit Town so you don't loose them. Left-click on the family's house and save them as household only.
The post below goes in more detail on how to save a household via Edit Town.
How To: Save a Household
http://www.carls-sims-4-guide.com/forum/index.php?topic=19060.msg356789#msg356789Once you have them in your household library, you can try them in a new save, in a new house to start, as a test to make sure things are working okay. If things are working, then you can move them. Providing their house or the lot their house is on is working properly.
The .bad file means that file is corrupted and can't be fixed. You can safely delete that file. If you have a .Sims3.backup file of your main save. Then your in luck, delete the .backup of the file For example:
Mainsave.sims3
Mainsave.sims3.backup
Again, delete the .backup part of the filename and you should be good to go. Launch the game, the save should be available at the Main menu to play.