Ubuntu – Why doesn’t Ctrl-C kill the Terminal itself

There is no difference between the kill command and the System Monitor's 'kill'. The System Monitor just gives some arguments to kill that I assume you don't know.

When you kill <process-number>, it's actually sending a signal to that process to cleanly exit. This is called a SIGTERM. In order to kill a process unconditionally, potentially causing data loss, you need to send a different signal:

kill -9 <process-number>

This sends a SIGKILL to the process, which immediately ends its execution. System Monitor uses kill -9 when you ask to 'kill' a process rather than simply 'close' it.

This answer on the SuperUser Stack Exchange should be of help.

When you close a terminal the terminal sends a SIGHUP signal to the shell; the shell, in turn, sends a SIGHUP signal to all its children process groups, which include backgrounded process groups;

How each single process will react to the signal is entirely up to the process: if the process didn't define a handler for the signal and tell the kernel (via some syscall such as signal() or sigaction()) that it wishes to handle it, the kernel executes the default handler for the signal, which in case of a SIGHUP signal consists in terminating the process.

However, when you run a command with sudo, the UID of the sudo process and its child process is set to 0 (root); in general, unless the UID of the process sending the signal is 0 (root) or the same as the target process, the kernel dismisses the signal (i.e.: a process can't send signals to a process owned by another user, unless the process sending the signal is owned by root); that's why an user-run process such as the Bash instance run by the terminal can't SIGHUP a sudo process and, ultimately, closing a terminal doesn't affect a process started with sudo.