No, this does not prevent the Task Manager application from closing. Task Manager may forcefully terminate a process that is not responding; he does not need permission to the application to close it, and it will not ask either. (See this answer for more information on the task manager and a comparison between the different ways to close the application.)
The only possible workaround is to have two processes, each configured to detect when the other is closed, and start a new instance. Of course, this still will not stop one of the processes from killing him, he will simply allow you to restart it. And this probably falls into the category of aggressive user behavior. If I resorted to using the task manager to close the application, I probably want it to disappear, no matter what you are as a programmer. And I guarantee to be insane if he continues to deploy new processes (he will also probably be crazy, this is my antivirus, because he had seen similar behavior before).
I recommend that you review your app design. If you need something that runs all the time in the background, you must create a Windows service. Of course, services do not have a user interface, and it looks like your application requires it. So even better, write your code on the defensive: save the state of the application so that it can be closed and restored at will. You should handle the case when the computer shuts down anyway, since it is difficult to cope with the fact that the application is closed?
As Microsoft Raymond Chen tells you, Windows does not have a mechanism for this, because no one could imagine an application as awesome as yours, which no one would want to close.
As for disabling the form closing window, the closing icon in the system / window menu, and pressing the Alt + F4 key, this is relatively simple. You need to override the CreateParams property form and set the CS_NOCLOSE window class style :
protected override CreateParams CreateParams { get { const int CS_NOCLOSE = 0x200; CreateParams cp = base.CreateParams; cp.ClassStyle |= CS_NOCLOSE; return cp; } }
Compile and run. You will get a form that looks like this (pay attention to the closed close button on the title bar and the absence of the "Close" menu item in the system / window menu):

Please note that when doing this you must really provide an alternative mechanism in your application interface to close the form. For example, in the form "master" from which this dialog was displayed.