In this context, that refers to the best. How OP can tell this is by assuming he should just work backwards from that looking for the first plausible candidate noun that precedes it (not far back, in this case).
As to exactly what the best means here, that's less clear-cut. It's probably easiest to just think of it as the greatest good for the greatest number. Analysing any more precisely is LitCrit, beyond the scope of ELL.
Most importantly, it certainly doesn't mean what was best for Firenze. When someone says they're doing something for the best, it's normally implied they're doing it for the benefit of others, not themselves. It's the same as acting with the best intentions.
Note that it's possible to "overanalyse" exactly what that refers to in such usages. For example, note that it would have been perfectly possible for Bane to have said...
"Acting for the best! What is that to do with us?"
...and it would be pointless debating whether that referred to acting [in that particular way], or to the best.
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.
Best Answer
This expression X has nothing to do with Y can mean two things (at least):
(That could be regarded as a harsh remark, but the phrase can also be used in a reassuring way, to say that something is not your fault:)
Back to what you were trying to say:
I guess I can sort of see how this might mean the time of day doesn't affect your English, but I'd probably not word it that way. Instead, I'd say something more along the lines of:
As for the other way around:
That one seems a bit harder to catch the meaning. I might word that like this: