Do European PhD programs have a qualifying exam, and if so, what form do they take?
Most central European universities don't traditionally have QEs. However, it seems to me as if it is getting more popular to have something after the first one or two years of your programme. Two data points:


In my current university, PhD students need to defend their thesis proposal after (maximum) two years. This includes writing their proposal, receiving written comments by two other (i.e., not their advisor) professors of the faculty, and presenting and defending their proposal in front of the entire faculty (our faculty is pretty small). Questions are asked in this defense, but not typically about material that is not directly linked to the proposal. In theory students can fail at this step, but is is very uncommon. The goal is rather to force students (and advisors) to have a clear goal of where the thesis is going early on, something that was historically a bit of a problem.
When I did my PhD, we did not really have any sort of entry exam or defense in my alma mater. However, since then, they have switched to a model not unlike what I explained above. The main difference is that proposals are only presented (there is no written document), and that only a small committee is responsible for giving feedback on the proposal (not the entire faculty). This defense has to be taken one year after start of the PhD. Failing this defense is again very uncommon.


Summary: the places I am well aware of don't have stressful QEs. Instead, we traditionally had pretty much nothing. Nowadays, many places have a proposal defense instead of a QE, but this is not a step that students typically have to be stressed out about.