The sense of the word 'live' in 'Why does that girl live?' is not encapsulated by the word रहना. You should use जीना instead.
रहना is 'live' as in 'stay' and जीना is 'live' as in 'be alive'.
So, वह लड़की क्यों रहती है doesn't really make any sense if the verb has no object. You'd need something like 'वह लड़की अमरीका में क्यों रहती है (Why does that girl live in the US?).
In the first sentence, we have an auxiliary verb 'does'. This is the verb that needs to be conjugated with the subject ('that girl'). Since the subject is singular, we are using the singular form 'does' and not the plural form 'do'. Due to the presence of the auxiliary, the main verb 'live' doesn't have to be conjugated with the subject (ie, it's always 'live' whatever may be the subject).
To make a very simplistic comparison, consider the sentence 'वे खा रहे है।' Only the auxiliary verb रहना is conjugated with the subject while the main verb remains खा whatever may be the gender and number of the subject.
In the second sentence, 'lives' is the only verb and it is being conjugated as singular.
Because it's incorrect English. You're saying: where is that girl live. The English auxiliary verb is 'to do' so the translation is: where does that girl live. (You could have used 'is' if the question had been where is that girl living.) Hope that answers your question ...