New to me, Q’s on nut/bridge fit and strings.
Instruments- discuss specific features, luthiers, instrument problems & questions
Glen, because you can tune the open strings to sound correct yet when you fret them at the 1st fret they sound sharp, will almost certainly mean the action at the nut is too high. Lowering the nut action so that the action is low and easy at that end of the fretboard will instantly get rid of this note sharpness problem in that area of the fretboard. The only thing that would prevent this simple solution from working is if the nut is incorrectly positioned and even more so if it's incorrectly positioned in combination with an incorrectly positioned saddle/bridge. All this is assuming the frets are correctly positioned, if they are not then that would complicate things a great deal.
Also, it's probably best to not rely too much on tuners for tuning, other than tuning one initial open note in order to set the pitch. After you get that open note accurate then tune all other notes via your ear taking into account the relationship of the subsequent notes to each other and to the initial note.
And one final fact, no fretted stringed instrument can be tuned perfectly for everything. Perfect tuning requires specific fret placement combined with specific tuning techniques, and those placement and techniques differ depending on the music played. In other words, what's perfectly in tune for one type of music can be quite imperfectly out of tune for other types of music. That's why throughout the cultures around the world there's many different and complex approaches to fret placement.
updated by @corvus: 01/15/20 09:06:11AM