Hi Y´all.
First of all, I would like to thank everybody for their great input/suggestions and the support from the forum.
Like I said, it would take a couple of "weeks" before I would got arround working on my CNC project again. But it finally happened

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Order of measures conducted, tried and applied, to sort the issue out is as follows:
1. Finished installing the linear glass scales on all three axis. (Amazon Chinese specials.)
2. FIne DRO adjustment procedure - as recommended and linked PDF by ShawnM
3. Lead screw compensation table via the installed linear glass scales.
Running the same test piece, as described in my initial post, I was constantly producing a part out spec by 0,1mm instead of 0,3mm. So not the perfect result I was looking for but at least it was consistent and better.
So what I did was to use tool compensation for that measly 0,1, and the results really supprised me. The part ended up within 0,01mm to 0,04mm. And I´m happy as can be. Not bad for a hunk of welded steel plate for a machine

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I´m pretty sure 0,1mm over/under size while using a professional machine like a DMG or Mazak would be an utter catastrophe, if you needed to use tool comp at such a "coarse" level.
But considering my machine consists of all the cheapest linear parts from Ebay/Amazon and is made out of steel, this is a great start. I can now use the machine to manufacture parts for its own upgrades.
Once again, thank you for your support. Any more advice is greatly appreciated.
Best regards:
P-J Johansson