Linux – ERROR: init: ttyS0 (/dev/ttyS0) main process (1612) terminated with status 1

By default, no, that's not allowed. Under Linux (from man 2 kill):

The only signals that can be sent to process ID 1, the init process, are those for which init has explicitly installed signal handlers. This is done to assure the system is not brought down accidentally.

Pid 1 (init) can decide to allow itself to be killed, in which case the "kill" is basically a request for it to shut itself down. This is one possible way to implement the halt command, though I'm not aware of any init that does that.

On a Mac, killing launchd (its init analogue) with signal 15 (SIGTERM) will immediately reboot the system, without bothering to shut down running programs cleanly. Killing it with the uncatchable signal 9 (SIGKILL) does nothing, showing that Mac's kill() semantics are the same as Linux's in this respect.

At the moment, I don't have a Linux box handy that I'm willing to experiment with, so the question of what Linux's init does with a SIGTERM will have to wait. And with init replacement projects like Upstart and Systemd being popular these days, the answer could be variable.

UPDATE: On Linux, init explicitly ignores SIGTERM, so it does nothing. @jsbillings has information on what Upstart and Systemd do.

First, “ancestor” isn't the same thing as “parent”. The ancestor can be the parent's parent's … parent's parent, and the kernel only keeps track of one level. However, when a process dies, its children are adopted by init, so you will see a lot of processes whose parent is 1 on a typical system.

Modern Linux systems additionally have a few processes that execute kernel code, but are managed as user processes, as far as scheduling is concerned. (They don't obey the usual memory management rules since they're running kernel code.) These processes are all spawned by kthreadd (it's the init of kernel threads). You can recognize them by their parent process ID (2) or, usually, by the fact that ps lists them with a name between square brackets or by the fact that /proc/2/exe (normally a symbolic link to the process executable) can't be read.

Processes 1 (init) and 2 (kthreadd) are created directly by the kernel at boot time, so they don't have a parent. The value 0 is used in their ppid field to indicate that. Think of 0 as meaning “the kernel itself” here.

Linux also has some facilities for the kernel to start user processes whose location is indicated via a sysctl parameter in certain circumstances. For example, the kernel can trigger module loading events (e.g. when new hardware is discovered, or when some network protocols are first used) by calling the program in the kernel.modprobe sysctl value. When a program dumps core, the kernel calls the program indicated by kernel.core_pattern if any.