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.
Some of the suggestions on this page are not grammatically valid. In fact, the phrase you propose is not grammatical. Since you don't provide context, I'll do my best to narrow things down for you.
There won't be anything happening to her
Is possible (i.e. is grammatical), though unlikely. This would probably be a sentence uttered by someone assuring another person of the safety of a female in the sense of
Don't worry. There won't be anything happening to her (Nothing will happen/is happening to her)
It could also be:
There won't be anything that happens to... (e.g. interfere with the process)
Using "happens to" in the sense of "to chance", giving us
There won't be anything that might interfere with the process.
Unless you provide more context (i.e. the article where you found it) I won't be able to help further. I can just confirm the phrase is not grammatical.
Best Answer
There is a discussion of something similar between the novelists Ford Madox Ford and Joseph Conrad. Ford was trying to explain to Conrad the nuanced difference between "without a penny" and "penniless".
I have nothing could mean "Everything has been taken from me." (e.g. King Lear)
*I don't have anything" could mean "At the moment, I don't have an answer, I don't have any money on me, I have no contraband on my person, etc etc"