1 - The printer has been removed from the Windows printer environment but still exists in the LocalServices client. Remove the printer there as well, or change the Windows printer name to an existing Windows printer.
2 - Printing is being done to a printer shared over the network. These printers start with \\ followed by the workstation name of the printer, followed by a \ with the name of the printer. In order to print to network shares, the LocalServices Windows service must also run under a network account. The print commands are sent by the Windows Service, and if it runs under the normal LocalSystem account, it does not have rights to network shares.