Frets 1-7 on E String considerably flat.

With Zero Fret, The nut won't make a difference. Are you saying that the Pressed down 12th fret E is intonated? Also, How high is your action? How much relief in the neck?

 

No. How could it?

The only thing impacted by nut slot (or zero fret) height is how much clearance there is over the 1st fret when you play the open string. Once you fret the string the nut is completely out of the equation.

 

Weird intonation problem on only one string could be caused by the string. Try a different E string if adjusting the action and intonation will not allow it to be intonated to your satisfaction.

 

The 12th fret harmonic matches 12th fretted?

Riis

 

Assuming a proper setup; going flat not only could be caused, but MUST be caused by a faulty string/string installation.

Before you change strings, ensure you tuned up to pitch and not down. You can check that by the simple expedient of checking the tune of your open and 12th again, without making any adjustments. If they are flat you may have tuned down to pitch. Take it flat and retune; if you over shoot the mark go flat and retune again. Rinse and repeat until you get it right.

 

Are you checking the note when the string is still moving?

You may expect variations in pitch until the string settles

 

i’d suspect a bad string, but also check that the string is actually resting on the zero fret and not being held up off it by the keeper nut thing.