Why child process still alive after parent process was killed in Linux?

No, when you kill a process alone, it will not kill the children.

You have to send the signal to the process group if you want all processes for a given group to receive the signal

For example, if your parent process id has the code 1234, you will have to specify the parentpid adding the symbol minus followed by your parent process id:

kill -9 -1234

Otherwise, orphans will be linked to init, as shown by your third screenshot (PPID of the child has become 1).

-bash: kill: (-123) - No such process

In an interactive Terminal.app session the foreground process group id number and background process group id number are different by design when job control/monitor mode is enabled. In other words, if you background a command in a job-control enabled Terminal.app session, the $! pid of the backgrounded process is in fact a new process group id number (pgid).

In a script having no job control enabled, however, this may not be the case! The pid of the backgrounded process may not be a new pgid but a normal pid! And this is, what causes the error message -bash: kill: (-123) - No such process, trying to kill a process group but only specifying a normal pid (instead of a pgid) to the kill command.

# the following code works in Terminal.app because $! == $pgid
{
sleep 100 &
IFS=" " read -r pgid <<EOF
$(ps -p $! -o pgid=)
EOF
echo $$ $! $pgid
sleep 10
kill -HUP -- -$!
#kill -HUP --  -${pgid}  # use in script
}