I think a robot has to have an arm. If it doesn't have an arm, it's not a robot -- it's just a machine. The whole robot could just be a single arm or the arm can just be one part. But it has to have an arm.

Yes. The dishwasher (DW) does not complete the entire dishwashing circuit/task, but certainly a major portion. That is, the DW can not load or unload itself generally, but it can sanitize, wash and dry among other tasks. Thus, meeting the general definition of a robot: carry out a complex set of instructions/actions automatically once programmed.

Not the OP, but I think the answer is definitely no.

Robots are not well defined, but I think at the minimum they imply the ability to be reconfigured for a large number of physical tasks that involve interactions with an environment.

My feeling is that the "robot" concept requires feedback loops and flexibility to adapt to a wide range of situations. As technology progresses, the required amount of magic required to warrant the title is a moving target.