What after exec() in ls command: Is the parent process printing the output to the console or the child?

ls will output what it has to output on its standard output. To do that, it calls the write system call, something like:

 write(1, "file1  file2...\n", 16)

(or more likely it calls libc functions like printf or fwrite that eventually do the write() system call)

It assumes that the file descriptor 1 (stdout by convention) was already opened and points to something. Actually, ls does check whether its file descriptor 1 points to a terminal or something else. If it doesn't point to a terminal, it does instead:

 write(1, "file1\nfile2...\n", 15)

That is, it writes one file per line when the output doesn't go to a terminal.

When you write:

 ls file1 file2

ls's file descriptor 1 will point to the same resource as the shell's fd 1 (so for instance, if it was an interactive shell started by xterm, that will point to the pseudo-terminal device controlled by xterm). The shell does nothing special, it's inherited upon fork and because the O_CLOEXEC flag is typically not set of the file descriptor, it is preserved upon execve.

If you write:

var=$(ls file1 file2)

The shell creates a pipe, and assignes the fd 1 of the child process to the writing end of that pipe and reads the other end of the pipe to fill the var variable.

It's not magically done upon exit, it's just done as part of the job of the process. It's independant from any other activity of the shell. ls is just another process with its fd 1 connected to some resource like the terminal while the shell is another process busy doing a waitpid().

What you may find though is that when stdout is not a terminal, ls buffers its output and would only call write() when enough data (enough being several kilobytes) have been accumulated or it's closing its output or it's exiting. So, in that regard, you'll find the write is done upon exiting, but only as part of the flushing of buffers that is being done by those I/O libraries.

Typically the parent process waits until the child process ends by calling waitpid. The parent process gets the PID of the process from fork.

This means the child never signals the parent process in any way that it exited or what happened. This is done by the system and not the child process.

If you are talking about the output of the program, the parent typically never receives the output of the child process unless it provided fds. This also means that the child process prints the output and not the parent process. The parent process just receives information about the state of the process (for more information see the macros in the waitpid manpage)

Strictly speaking neither of parent and child do the actual output to the screen; instead the operating system kernel (specifically the tty driver) does the output. The processes only send the data to a file descriptor.

Also, instead of speculating strace -f /bin/bash "ls;exit" clearly shows who is writing what to the file descriptors (BASH seems to optimize "-c ls" by not forking, so I used the ";exit" afterwards), like this (bash uses clone, not fork):

...
clone(child_stack=0, flags=CLONE_CHILD_CLEARTID|CLONE_CHILD_SETTID|SIGCHLD, child_tidptr=0x7f9814b5d9d0) = 30029
Process 30029 attached
[pid 30028] rt_sigprocmask(SIG_SETMASK, [],  <unfinished ...>
[pid 30029] rt_sigprocmask(SIG_SETMASK, [],  <unfinished ...>
...
[pid 30028] wait4(-1,  <unfinished ...>
...
[pid 30029] execve("/usr/bin/ls", ["ls"], [/* 72 vars */]) = 0
...
[pid 30029] write(1, "00708022-PTF-26962\nCASE00280714\n"..., 18500708022-PTF-26962) = 185
...
[pid 30029] exit_group(0)               = ?
[pid 30029] +++ exited with 0 +++
<... wait4 resumed> [{WIFEXITED(s) && WEXITSTATUS(s) == 0}], 0, NULL) = 30029
...

So you see that actually PID 30029 (the child) writes out the file names, exits, and then the parent continues (after wait4).

If exec(3) succeeds, the program which called it is no more. It is replaced by the program execed. The new program running inherits the environment, in particular the open files, of the original.