The windows and overall shell play a bigger part on efficiency than a door ever will. With that, by having it on the north it will mainly be the coolest section though I would recommend a covered patio or walk in area no matter where it is placed.
As for the pool, have you considered a courtyard with walls and a pergola roof area? Think Santa Fe or Mission style.
For more information:
Read "Which way should we orient our green home in Boca Raton, FL?" a Q&A answered by Randy Potter.