You can use the "Remove Error Markers" keybinding, also available as menu item in the Document menu.
Yes, that's right. But a shortcut would be much better,
"Remove Error Markers" keybinding is a keybinding! Find it in the Preferences dialog under Keybindings in the "Document" section.
I mean, why you offer a toggle-version for "Check while Typing", but not for "Run Spell Check"?
As said above, because "Run Spell Check" is an action to execute and not a state that could be toggled.