Re: Tuning On NEW LP
Your tuning issue is absolutely the intonation being off. LP's, since they have a few things working against them, like a shorter scale and increased headstock angle, makes it VERY necessary for the intonation to be 100% perfect.
First, make sure the action and truss rod adjustment is exactly where you want it.
Then, tune the guitar with new strings on it. Make sure the open notes are dead center on the tuner, then make sure every note from the 10th fret to the 20th fret on all strings is DEAD CENTER on the tuner, as well. If they are a hair flat, move the saddle toward the neck. If those notes read a tad sharp, move the saddle away from the neck. Do this until all your open notes and all the upper register notes read dead center on a good tuner. You'll notice that the guitar will stay in tune nicely, and all of your chords across the neck will sound perfect.
If the nut is cut slightly high, which is typical of Gibson, forcing notes to bend slightly sharp when fretted at the 1st and 2nd fret, you need to have a tech run a file through those slots to bring them down. If all these issues are addressed, your guitar will play beautifully. Add a C-5/59 nickelcovered, and you're THERE!