PHP: Why isn't exec() returning output?

You should redirect stderr to stdout.

To do that, change your exec() call like this:

exec("ping -c 1 $domain_bad 2>&1", $output, $return_var);

More info about 2>&1 meaning here.

If the answer above could not solve your problem, maybe exec() is disabled. You can try to check php.ini file at disable_functions line.