Hello.
How can I execute a command as server? Because if I start a new world with "Allow Cheats"=false, the player has no permission to execute a quest command which should unlock his inventory.

and why is the command output visible in the Chat? Can I disable this?
And is there an config option for executiong commands as server?

This is not logging info for server operators, but rather the direct feedback to command executors, so the problem is entirely in how command rewards are resolved.

So for your second question, you probably see the executed commands because it is enabled in the server.properties and you're operator