"Are you staying here for many days?"
Translation:¿Te quedas aquí por muchos días?
Having plans 'for the summer' is para, but staying 'for many days' is por. Now I'm lost. I though 'para' meant 'for' more in terms of 'in order to' ???
Quedarse is the infinitive that means 'to stay' or 'to remain'. Your sentence has no subject, and it would translate as 'to stay here for many days?', but no one would know who or what is staying/remaining. Also, quedarse is a reflexive verb, which means it needs a reflexive pronoun to go with it, e.g. me quedo, te quedas, se queda, nos quedamos etc.