Hi Jarro, I'm liking the look of your guitar, good work.
I'm no expert, but I'll share what I would look at first and hopefully it will help
1. The noise problem. I think you may have an issue with your shielding. Since it is quiet when you are touching it, I think you are picking up noise that is external to the guitar. Simon made a few comments that are worth checking a few posts back. Firstly, make sure you have connected your shielding to earth, and secondly, make sure it goes high enough to connect to the shielding on the back of the scratch plate. If the shielding is working, it might address the noise.
2. I don't think I am following you completely in intonation, but it sounds like you are having issues with tuning of the first 5 frets. There might not be much you can do here, and let me explain my thinking. Intonation is normally set by comparing the 12th fret tuning to open tuning. If you have done this, and are finding the first 5 frets are out of tune, it might be worth re-checking the scale length and expected fret positions. I use the stewmac fret calculator, and you can see if the nut seems to be correctly positioned. My nut was leaned forward slightly and I filed it backwards ever so slightly to correct the scale length. Keep in mind though, guitars on the whole are not perfectly timed instruments and this is part of their charm.
3. Pickup height it a bit subjective, but as a rule of thumb, it is safer to be further away than too close. The closer you get, the more effect the magnets have on the string vibrations, and they can reduce sustain and affect the tone. Further away can result in a somewhat thinner tone, but there is more leeway on the far side of the equation
4. Open string buzz normally comes from not enough neck relief. 0.01mm over 300mm is really, really small (it is essentially flat). Maybe try adding a little more neck relief by backing off the truss rod a little. it's a quick adjustment and can be undone if necessary.
I hope this helps, it's worth fine tuning and adjusting thing, it takes a little time but it is well worth it for making a guitar feel great to play.