the stress often looks quite noisy/wavy with over-/undershoots. This might looks incorrect, but is a result of the finite element solution only being continuous between cells, which means that its derivatives, and therefore stress and strain, have jumps between cells.
I created a pull request to change the default in ASPECT to output stress averaged per cell here: https://github.com/geodynamics/aspect/pull/3478
If you want, you can take a look at it to see if this makes your output look more reasonable.