Hello,
We do not at present log these anywhere other than the PuTTy window. In principle you could get it to write to a file in the Python back end, but this had not been implemented.
Usually what I do is copy-paste or screenshot the putty window as needed.
It sounds based on your description that the output pump (the physical items) is not performing so well. If it is making a bad seal on the tube then indeed it could fail to remove liquid and it can overflow. It could be a manufacturing defect – in the first case I’d try swapping it out with pump 3 or 4 on the board and see if that fixes it.