Why is "na to" in the first clause of the sentence. Would "Jsi dost stará, abys to věděla." be correct?
It is, more or less, an idiom. You can skip it but it is commonly used here. It means "for it". "You are old enough for it, so you should know better" would be a loose translation.
Why would abyste be wrong here? It gave me the option between abys and abyste, and I don't understand.
This sentence seems to be disabled for the czech side, so not sure where it popped up at you. Nevertheless, yes, abyste should be fine, but do not forget that in such a case, the first part of the sentence also has to be formal and use JSTE dost stará...
'jsi' is the informal singular you form, so you need the informal singular 'bys' to match it.