If you wanted to be a pain, this could also be translated as "I swim and she...does nothing" :)
"I swim and she doesn't." Without further context, I believe this should be correct, am I wrong?
If the sentence was "I swim and she swims" on the team, that would be different.
In that case you would translate it as 'I swim and she nothing'. If you wanted to say it your way you would say 'Eu nado e ela faz nada' or 'Eu nado e ela não'
There's a different between nada. You can use "nada" in the case of "she swims" because the pronoun is ela. And there's nada as nothing or anything.