Usages vary, but I'd say 'a runny nose' and 'a stuffed up nose'. I know that the term 'stuffy nose' exists, but I think that 'stuffed up nose' is far clearer - it means that your nose is blocked and that you're having trouble breathing. As for the difference between 'She's got...' and 'She has..' - there isn't one. They mean exactly the same thing. 'Runny nose' and so on are informal phrases, so it's fine to use the slightly more informal construction 'She's got'. Note that we rarely say 'She has got' - this form nearly always uses the contraction 'She's'.

Both are correct,Milka there is no difference in the meaning.