PG: If the neck (arm) is straight and true, I don't know why then the strings are getting higher off the finger board (fret board). That should then be causing you to sound 'out of tune' a bit when you get to the 5-9 fret area. If you capo and play at those frets, then you likely have been retuning a bit after you position the capo. Maybe evenn a bit more than a bit.

I ran into a similar problem (my D 28 is now about forty or forty-one years old) when I played in the E position at the seventh fret. I found there was a slight warp in the neck. To check for that, hold a string down at the 1st fret and the 12th fret. If there is clearance between the frets and the strings when you look at the machine sideways, there's a warp. Check all six strings, one at a time. Also, pick up the guitar and hold it with your left hand at the head and your right hand on the back, just like you were going to put a fiddle under yer chin. Look down the neck towards your left hand. If you have any 'torsion' warping, that will show it to you.

I don't know anything about pickups (other than Fords).