Learn English – cannot do anything or can do nothing

Your b) versions are ungrammatical in standard use because they employ double negatives.

I cannot say nothing else would only be used with emphasis on the 'nothing' to assert that you do in fact have something else to say.

I cannot say anything else would require a just before anything and would emphasize anything to assert that the something else you have to say isn't just anything you say to be part of the conversation but is highly specific or relevant to the topic at hand.

If you want to use 'nothing' in these contexts you must delete the 'not':

I can say nothing else because I noticed nothing else.

Even in non-standard use, where double negatives are sometimes employed for additional emphasis, you are unlikely to encounter them in the forms you offer; the negated verbs would be contracted:

I can't say nothin else cause I di'n't see nothin else.

In my part of the country you wouldn't say even that; you'd say:

I can't say nothin else cause I ain't seen nothin else.

In these sentences, "do they really" essentially means "is it really true that they". If you rewrite the sentences like that, it should become clear which is correct:

Is it really true that they have nothing in common?

Is it really true that they have anything in common?

Obviously the version with nothing is correct (the statement you are curious about is "They have nothing in common"); the version with anything doesn't make any sense.

But what you might be interested in is a similar question using anything:

Do they have anything in common?

This asks simply that; do they have anything in common? By itself, there is no implication that the speaker thinks they do or don't (but it's likely that intonation would make this clear.) Since your question referred to confirming an existing suspicion, you might consider a conversation like this.

Person A: Do they have anything in common?

Person B: They both know the sky is blue?

Person C: Oh, come on. Do they really have nothing in common?

You might also consider this common variant, as mentioned by StoneyB in comments:

Don't they have anything in common?

Note that anything is stressed, and don't is used rather than do. The implication is that the speaker is surprised to learn that the people supposedly don't have anything in common, and is checking that this assumption is true.