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I'm kind of confused by Feynmann diagrams where a stray gluon flies off. What happens to it then? Where does it go?

It goes into a vertex involving some other particle with the appropriate "color" charge. That part is not depicted because it is not relevant to the process under study, as in "and then it goes off to die somewhere, I don't care where or how".

That's just a part of a bigger diagram. Due to color confinement you can't just have one gluon.

They have a diagram that emits a single gluon in the head of every particle physics article in Wikipedia. Is that a make-believe diagram? I couldn't expect that from them.

It is not a make-believe diagram. It is a completely valid diagram, but it i just part of something bigger. Also the two quarks in the diagram need to ultimately couple to something else. The "in" and "out" final states must be colorless. QCD is a extremely complex theory, one usually computes a piece of the puzzle and tries to extract physical information from that.

Just to add to this: The part that comes this initial gluon emission (hadronization) is not very well understood since at this length scale the perturbative (i.e., feynman diagram) approach to QCD breaks down. See e.g. http://www.quantumdiaries.org/2010/12/11/when-feynman-diagra...

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