See kill -l for a full list. All the kill command real does is send a signal to the target process(es). The other commands you list are variations on this basic theme -- read their man pages for more details. How the process responds depends on how (and if) it's set up to interpret that signal. The only signals processes can't have handlers for are SIGKILL (number 9, hence the infamous kill -9) and SIGSTOP (forcibly suspends process execution). Many of the other signals date back to days of yore when folks interacted with computers via teletypes and dial-in lines (e.g. SIGHUP used to mean that the terminal line had physically hung up, but now processes often interpret this to mean restart execution or reopen log files).
The only signals end users need to worry about are generally SIGINT, SIGKILL, SIGTSTP, and SIGSTOP. The first to kill a process (SIGKILL forcibly, without giving a process a chance to handle the signal, so it's not a good idea to do that unless you have no other choice), e.g. in most cases pressing CTRL+C sends SIGINT to the foreground process. Likewise the latter two stop a process (SIGSTOP forcibly). SIGTSTP is what CTRL+Z sends top suspend a process, for example.
So it's a little bit complicated ... kill sends a signal (which may or may not terminate the process). Within limits, what the process does with that signal can be defined by the program.
Fun facts: init cannot be killed, even with kill -9. Processes in state D (I/O wait) will not be signaled until their state changes, as it's unsafe to terminate processes with locks on I/O buffers.