No. Once you have scheduled a backup task, you can quit CCC because a "helper" application will be running quietly in the background, waiting to start your task. Scheduled tasks also load automatically when you restart your computer, so you don't have to launch CCC again unless you want to change how your task is scheduled.
The scheduled task will run whether someone is logged in to the machine or not. If no one is logged in to the machine, CCC will run in the background without a user interface. CCC logs information to /Library/Logs/CCC.log, you can use that log file to monitor CCC progress (e.g. logged in remotely via SSH).
If you enable the option to "Wake/power up the system for this task", CCC will wake or turn on your computer when your scheduled task should run. Without that setting, if a backup task is scheduled to run while your computer is sleeping, CCC will run that task immediately upon wake (CCC waits 20 seconds to allow your machine to establish network connectivity and spin up hard drives).
If your Mac is a laptop, note that CCC will not be able to wake the system if the lid is closed. The only exception to this rule is when the laptop is connected to a display that remains turned on (it's OK for the display to sleep). The laptop must receive an active signal from the display or it will immediately go back to sleep after the scheduled wake time.
Nothing runs when your computer is turned off. Starting in CCC 3.4.4, however, you can configure CCC to turn your computer on at the time that the task is configured to run. With or without that setting, however, CCC will automatically run a scheduled task shortly after your computer starts up if it has missed a scheduled task interval. So, for example, suppose you have a scheduled task configured to run at 9PM, but you turn off your computer at 8PM. When you turn on your computer in the morning, CCC will run that scheduled task because the 9PM interval was missed. If it is inconvenient to run the task upon startup, you can click the Stop button to stop the task, then click the Skip button to skip the missed interval. CCC will then resume a normal schedule for that particular task.
If your backup hard drive is unavailable when the task is scheduled to occur, CCC will report an error that the volume is not available and ask you to skip or defer the task. If you choose to skip the task, CCC will immediately run the backup when you reattach your hard drive. If you do not attach the hard drive to your computer before the next scheduled run time, CCC will prompt you again that the destination volume is missing.
Yes, you can stop the backup task at any time. The next time you run the backup task, CCC will copy only the files that have changed or were missed since the last backup task.
Carbon Copy Cloner Scheduled tasks are managed by launchd. Each scheduled task exists in the form of a property list file stored in /Library/LaunchDaemons. When these LaunchDaemons are loaded, an instance of "ccc_helper.app" is launched and configured with the scheduled task settings. While the Scheduler interface in CCC should be the primary means of configuring scheduled tasks, you can effectively deschedule a scheduled task by removing its configuration plist and rebooting your Macintosh.
The scheduled task helper application, ccc_helper, runs with elevated privileges as the System Administrator (root) user. This allows scheduled tasks to run when no one is logged in to the system, and also allows the tasks to load consistently when your computer is started. That the application runs as a different user than the one you log in as, though, causes its window to be treated differently from other applications' windows. Most notably, the application cannot be minimized to the Dock. If it could be minimized, it would be minimized to the System Administrator's instance of the Dock, which would not be visible to you.
Starting in CCC 3.4.4, there is an option to "Display a progress window during the backup task". This setting is enabled by default, but you can uncheck this option to hide the scheduled task helper application's window. You will find this setting in the "Settings" tab of the Scheduled Tasks window.
The configurations available in the Scheduler window can be configured at any time. You cannot, however, modify the task configuration (e.g. the source and destination volumes specified, cloning options, selected items to be copied, etc.) — yet. If you would like to make changes to the task configuration, delete the current scheduled task, then create a new task with different settings applied in the main window of CCC. Modification of scheduled task configuration is planned for a future release of CCC.