Which personal pronoun correctly completes the sentence? what is its use?
dad asked joe and to put away the dishes.
a. i; indirect object
b. me; indirect object
c. i; direct object
d. me; direct object

This would be a lot easier to recognize if Joe and the dishes were not involved.
Then it would simply say:
                                           Dad asked me.

'Dad' is the subject.  'Asked' is the verb. And 'me' is the direct object of the verb.

None of that changes when Joe and the dishes enter the scene.

Dad would not ask ' I ', and he certainly didn't ask the dishes.

It's D, "me; direct object."