What processes cause the collapse of a wavefunction and break entanglement?

Since you already talk about Stern-Gerlach, I suspect that the focus of your question is more about at which point in existing experimental techniques the collapse occurs, and not about learning existing techniques. In Stern-Gerlach that would be the deflection, not the screen, because this is where the spin value gets to be determined. If I got the question right, then the general answer is "at the point in the experiment where the studied property gets a specific value and superposition ends".

Also: Measuring is interacting with the system under study. There is nothing special putting measurement apart from any other physical process. This means that any interaction of the original system with anything else in the universe will break the wavefunction, preparing a new state.

I think the most concise (and entertaining) answer was the first comment (source) in the first of your links:

Basically, for observations to happen, there has to be an interaction between particles, or as the post put it less/more(?) eloquently, whenever a physicist says "observe", mentally replace it with "hit with sh*t.