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I created this mesh and checked it for Scale, Doubles, Merge by distance and Face Orientation.
The mesh is 1 whole.
In my opinion it is, first make the mesh and only then Solidify, but when I solidify this mesh I get the strangest results.
What am I doing wrong" ? 
I want to solidify the entire mesh to the outside and 2mm thick.
To elaborate a little bit more on Crantisz' answer...
Here's a reproduction of a part of your geometry, using solidify without fixed thickness:
Light blue area represents what you want and what fixed thickness will achieve. Notice how the solidify reaches exactly the tips of the blue lines - this is because, well, I adjusted the thickness for that to be the case, but I could achieve this with just one value of thickness for whole geometry. This is because the simple solidify modifier simply extends the vertices in a direction of vertex normals. The blue line coming from the bottom-left vertex doesn't reach the opposing corner of the red square - this is because in order to do so, it would need a length of specified thickness (which is the length of each side of the red square) multiplied by $\sqrt{2}$ (because a diagonal of a square has length $a\sqrt{2}$, where $a$ is the length of a side).
It seems even thickness simply calculates how much to increase thickness locally to deal with that. It does not, however, deal with "wrong" angles of vertex normals. An "incorrect" geometry (non-manifold, ngons) may produce such angles; here's an example from your geometry, marked red:
How can a vertex have a normal?
Keep in mind, the documentation says this about the Solidify Modifier:
Known Limitations
Even Thickness
Solidify thickness is an approximation. While Even Thickness and High Quality Normals should yield good results, the final wall thickness is not guaranteed and may vary depending on the mesh topology. Especially for vertices with more than three adjacent faces.
In order to maintain a precise wall thickness in every case, we would need to add/remove faces on the offset shell, something this modifier does not do since this would add a lot of complexity. The best option to preserve wall thickness is complex mode with constraints thickness mode, but it is also not guaranteed to work perfect in every case.
And finally a comparison between simple mode with even thickness vs complex mode for something similar to your topology:
Solidify is not a perfect algorithm, which can handle every model.
For example, the things that you have pointed. If you solidify a plane, it goes like this (let's name it "plane"):
If you solidify an edge, it increases the thickness at the corner to save distance to face at same level (let's name it "edge"):
problem started then you try to merge "plane" with "edge"
Small extrude to keep "edge" solves problems like this:
