Thanks @Jerbear114. I got this guitar about 20 years ago, then stopped playing, so it hasn't been played much but it has had strings on it all that time. I don't think it was set up well (if at all) from the factory since it always had buzzing frets and poor intonation. I recently got it out and tried to set it up on my own and now the action is great but intonation is still off. Half of the strings intonate correctly, and half are still sharp. And yes, as you can see, even the ones that sound correct have the saddles all the way down, which indicates that something isn't right.
I have already done the following:
Put on new strings (9-42's)
Lower the action. With the 1st fret pressed and measured at the 15th fret:
the height of the 6th string 1.93mm (thickness of an EU 10 cent coin),
the height of the 1st string is about 1mm (thickness of a 42 gauge string)
Adjusted the truss rod so that, with the 1st and 15th frets down, the height of the string over the 7th fret is between 0.009" and 0.011" (I used guitar strings to measure, the 9 gauge can pass between the fret and string but the 11 gauge can not.)
Ensured that the bridge doesn't appear to be leaning forward
Raised the tail piece in an attempt to lower tension on the strings
Adjusted pickup heights:
Bridge to both E strings: 2.4mm (thickness of an EU 50 cent coin)
Neck to both E strings: 3.2mm (thickness of an EU 1 cent coin stacked on a US penny)
The next things I'm going to try:
Put on heavier strings, maybe 10-46's
Turn around all the saddles so I can move them back a bit more. (It seems like that wouldn't really be fixing the problem though, right?)
Double-check the nut measurements (I really hope this isn't the problem since I can't fix that)
Ask for help here 😀
Buy a wider bridge so I can slide the saddles back even further
If you or anyone else has some advice now at this time, I'm all ears!