Travel with Pets - Madrid airport in transit with a dog

I (US citizen) am arriving at Madrid T4s and departing (2 1/2 hours later) for Paris from T4. My questions: 1) Do I need an in transit visa (I've read that to go from non-Schengen T4s to Schengen T4 requires going through passport control, 2)Is there an outdoor area at T4 or T4s where I take my dog to relieve herself???

Thanks!