Error rate of AdaBoost weak learner always bigger than 0.5?
As far as I understand, weak learners of AdaBoost should never yield a error rate > 0.5


This is only true for binary classification problems.  The simplest possible classifier would pick the majority class, ensuring that accuracy is always >= 0.5
For a problem with $N$ classes, the worst error rate for a weak learner will be $\frac{N-1}{N}$.


  Is it possible that my threshold for the error rate instead is 0.9 (accuracy 0.1), as I have 10 classes and literature mostly focuses on binary cases?


Yep that's exactly right.  Your weak learners will have, at worst, an error rate of 0.9.  Unsurprisingly, most of your learners have a better error rate better than random guessing, which explains the good performance of your ensemble.

Edit:


  Also the trees get worse from iteration to iteration


This is a common feature in boosting algorithms.  Each iteration, training samples are reweighted such that the mistakes from previous iterations are given more importance.  The net effect is that the learners in later iterations are specialized for classifying the examples that earlier learners got wrong.  Since later learners are specialized for a few hard-to-classify examples, they will have worse performance on the entire dataset.