Hi Micha,
> Or here - additional polygons at the lower edge ...
Like I mentioned earlier in this thread, it can help for you to switch to Output:N-gons which more clearly shows you where the polygons are coming from.
So for example, with the parameters that you show there if you switch to Output:N-gons, you can see this (viewing from the bottom looking upwards):
Notice how the ngons are not aligned?
In general if you are confused about why you are getting triangles in a particular structure, I would encourage you to switch to Output:N-gons to see how the n-gons are formed. Any place where an n-gon is formed with more than 4 points in it, will end up with that being converted into triangles when you have Output:Quads & Triangles mode.
Now as to why they are divided differently - that is pretty simple, it is because those surfaces have different lengths, here I have discarded the other ones, and you can see that the veritcal one is trimmed back and is not as long as the bottom one:
Because they are of different starting lengths, the longer one gets something like one extra division on it when they get their uniform grids.
I can understand that it seems like they should align like a voxel grid would align - unfortunately it is not feasible for the mesher to use that kind of "intersect with world grid" method because it would only work well on things that happened to be made of planes that were aligned with the world axes. It would produce strange results on anything that was angled.
One of the things that you are asking a lot about are special parameters or things that would help out for a specific model. I can understand that, but just keep in mind that there are a really wide variety of models that the mesher needs to be able to process, there are often all kinds of things that would work great for just one particular kind of model, but that is just not generally practical. Those kinds of things would help you out in one situation and then be really weird and cause problems in a thousand other situations...
- Michael