Is the following a correct sentence:

Martha and him ate the sandwich.

No.

The easiest way to see if it's correct is to drop the first two words.

Would you say? Him ate the sandwich?

Of course not. But you would say: He ate the sandwich.

Therefore the correct sentence is:

Martha and he ate the sandwich.

Martha and he are the subjects. Since him is an object pronoun, we can't use it as a subject.