Finder and Trash icons in the Dock
The Dock Contextual menu, which you can show for each item in the Dock by right-clicking or cmd+left-clicking the item, contains Remove from Dock menu item (in the Options submenu), which enables you to remove specific item from the Dock. This is not the case for Finder and Trash items by default, which have a special treatment. Path Finder has two preferences settings, which enables you to show this menu item even for Finder and Trash. These settings are separate, so Finder and Trash are treated independently and they are located in the Finder tab of Path Finder preferences panel, like shown in the picture below.
It is important to know that Path Finder only enables Remove from Dock menu items to appear for Finder and Trash as well, it does not control or influence actions invoked by those menu items in any way. Those actions are controlled and executed by the Dock itself. Hence, Path Finder cannot control and influence the following:
Appearance of Finder and Trash icons every time you login to your account or restart your computer even if you removed them before.
Trash's menu item Open showing the contents of the trash in Finder instead of Path Finder.
The two above are hard-wired into the Dock behaviour and Path Finder cannot change that. Also, if you decided to quit Apple's Finder (using Quit Finder button or Quit Finder when Path Finder launches in the picture above), anything you try to do with the Trash icon in the Dock will not work. The Trash icon in the Dock is hard-wired to Finder and it requires it to be running. In such case, you can decide to show Path Finder's Trash icon in the Desktop (in Path Finder preferences panel select Features tab, section Desktop and check Show desktop trash can checkbox) and use it instead.