Solved for Windows 10; Probably also works for Windows 8
In Windows 7, I could easily start Task Manager at login by simply placing a shortcut to taskmgr.exe in my Startup folder and configuring the shortcut to start the program minimized. This method does not work in Windows 10 and I suspect it does not work in Windows 8/8.1 either (I no longer have Windows 8 or 8.1).
What I wanted is to have Task Manager start minimized automatically at login and to be placed in the System Tray without showing up on the Task Bar.
To solve this problem, I created a short custom Auto Hot Key (AHK) script, compiled it and placed a shortcut to the compiled script in my Startup folder.
In Task Manager, it is also important to enable the option to "Hide When Minimized".
I have attached a zip file to this post; it contains the source AHK script and the compiled version (.exe). You only really need the compiled version (.exe). The source AHK script is only needed if you want to make your own modifications (you will then also need to get the Auto Hot Key program itself and/or SciTE4AutoHotkey [highly recommended if you are going to work with AHK scripts]).
Let me know how this works for you.