PLC with or without PC
Can the PLC run without an external PC, and if so at what cost? (apart from the PC price, obviously).


Absolutely a PLC will run without a PC. The costs vary greatly depending on features (# of I/O, Redundant processors, etc. etc.)

That being said, the only thing the PLC would be able to control is the attached I/O (Contactors, Relays, Alarm Lights, etc.).

You would not be able to do any programming of the PLC. You could have a lap top computer (for instance) with a programming cable (usually USB or Ethernet) that you could use to program as needed.