Log world events

tMod is dedicated server software that allows you to write plug-ins that hook the events that you describe, their forums have some tutorials that get you started. Given that you directly log the hooks and only need single conditions on perhaps the mobs being killed, it shouldn't become to complex...

You can't really do it unless you write your own server software or wrapper (or find a tool someone else already wrote; I'm aware of none).

What you could do is redirect the server output to a file and then parse that (TerrariaServer 1> events.txt), but out of your list, that pretty much only includes player log-ins.