I need a special PERSISTENT checkbox type that keeps a file/folder checked even after the folder is reloaded. For example, I need a folder checkbox to permanently mark its COMPLETED state, e.g. to show whether a sale has been completed:
Use Status Icons for that. They should be in the right-click menu, as well as the Properties button's menu on the toolbar.
Thanks. I have clicked the Checked Status menu item:
But the Status icon does not show up in the LIST VIEW!
Activate it by right clicking columns.
Use Details mode. List view inherently does not show as much information and is only useful if you just want a more compact list of files. (Even then, I'm not sure List view is really useful. 
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(You can also see status icons in Power mode, and Tiles mode if you add {status} to the tile definition. {status} also works in the info-tip definition for all modes.)
If you want to stick with List mode, using labels that change the color or font of the file/folder names or which change their icons is your best bet. They work similar to status icons but change what's already being displayed instead of adding something extra.
Thanks. The Tiles section in Preferences does not have a "tile definition":
Maybe it wouldn't be a bad idea to have an option to auto-activate the column, in case there are status icons (if that's not already the case).
It is auto activated when you apply an icon, but the column doesn’t exist in List mode. (No columns exist in List mode except Name.)
Alternatively, you could also apply statuses by assigning them a hotkey. I have ctrl + space set to apply the "complete" status and ctrl+alt+space to apply the "flag" status.
Yes, you're right of course. I think it was different in older version though. But it makes a lot of sense.
I have SIX columns in List mode as default:
You have six things you can sort by there, but List mode won’t show the columns themselves.