I observed that a lot of admin_level boundaries (2,3,4,5 and sometimes
  6,7,8,9,10,11) are not useable because they are not completely
  contained in the tile data. The polygons are closed automatically but
  this is only a good guess in which direction they have to be
  closed. The probability is quite high that they are closed in the
  wrong direction.

I think this is a fundamental issue with splitting and areas.  Martin's
suggestion about extracting boundaries and using them on the side makes
sense, but I would say that the issue is about all large areas.

I wonder if it's possible to evaluate areas (and things like boundaries
which are semantically areas even if they are technically closed linear
features) for overlaping with tiles, and to then replace the area by
multiple areas projected onto tile boundaries.

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