Why doesn't codec tweak tool have taskbar entry?

There is no way to know there are updates because update checker window always gets covered by other windows.
Also why can't installer close Codectweaktool.exe by itself?

It shows in background because people don't want it to interfere with their work/gaming/etc. You should see it after closing windows.

The installer is capable of terminating the exe. Isn't that working for you? It first asks to close it yourself.