Why can't I delete?

The first one you are clearly in some subfolder of Applications.

Applications can only be managed by administrators, and software installers.

Only User folders are technically both read and write under all environments, as long as the user in question has permission to write to that folder.

Files that get dumped in applications by accident, can be moved by an administrator while booted into safe mode (holding the shift key). Be mindful once you empty the trash, it is gone for good unless you have a backup somewhere else.

Move to trash, leaves it in trash. You still have a chance to delete it from trash, or move it to another folder until you either delete it from trash or empty from trash.

Empty from trash gets rid of the programmatic pointer to the file, which instantly can be overwritten if you have little space on your system. Rarely, you can hard power down your computer, and find the item you had just emptied back in the trash, on next startup. Note, a software shut down does not allow for this.

Empty trash usually (NO GUARANTEE) is recoverable if you immediately shut down your computer and use a data recovery software.