Hmm, If a pitch pipe says you're out of tune, but your fifth fret implies otherwise, I'd say there's a problem with your guitar's intonation, meaning it's in tune when it's open, but when you fret, the pitch gets a little wayward.

My first acoustic had the same problem. So, I tuned my guitar based on the song I needed to play. If I were playing a song in E, I would fret an E major chord, then use a piano or my ear to make sure all the notes were right.