"She is just as fit as her father."
Translation:Sie ist genauso fit wie ihr Vater.
Basically, both "sie" and "ihr Vater" are subjects of the verb in the comparison, and so we could write this as "sie ist genauso fit wie ihr Vater ist", but the second "ist" is redundant, as it's obvious from context, but even though we leave out the verb, ihr Vater is still nominative (this is pretty much always the case with comparisons - the two things being compared are in the same case).
The first comment here gives a more detailed explanation if you need it :)
No, this sentence is about two subjects on the same level as each other, you have to use "wie". This is similar in English, where you have to use "as," not "than"; saying "sie ist genauso fit als ihr Vater" sounds just as wrong as "she is just as fit than her father". You can't just mix the two.
größer ALS- bigger THAN
so groß WIE- as big AS