Install ClickOnce without running

I guess you could fake it. Introduce an "IsInstalled" boolean property, defaulted to false. Then in Program.cs, change your Main() method to look like this:

[STAThread]
static void Main()
{
    Application.EnableVisualStyles();
    Application.SetCompatibleTextRenderingDefault(false);

    if (!Properties.Settings.Default.IsInstalled)
    {
        Properties.Settings.Default.IsInstalled = true;
        Properties.Settings.Default.Save();

        MessageBox.Show("Install Complete");
        return;
    }

    Application.Run(new Form1());
}

So now when the app is first installed, it checks that property and simply displays a message to the user and then quits.

If you wanted to get tricky then you could look at parsing the Activation URI for the deployment and have a URI parameter which specifies whether the program should run when it's first installed or just close silently.