Hi Lennard,

the intonation should be adjustable at the bridge, there will be adjuster screws that should allow you to adjust the bridge saddles. Forward or backward to balance out the sharpness or flatness of your intonation on each string.

The nut groove depth and nut height should only influence your string action and potential buzz.

Your truss rod will only straighten or bow your neck, which won’t give you a great deal of assistance with your intonation.

Shimming the neck should only be done if you have a serious action or fretboard height/angle issue.

what kind of bridge is on it, Tremolo or Hard tail?