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Search around and you will see that they are very common and made in literally thousands of different shapes and sizes. The photo you posted is merely one example.

30...

Might try lubing the slots with graphite from a pencil.

Nope, you make the slot tapered and it rests in the bottom of the slot if it's the right shape. Kind of like how you can put a tennis ball in any sized bowl, or pour sand through a funnel. You just...

A cleat is a small sliver of wood glued across a crack to reinforce it.

Sure, but the separation in this case is most likely not cleated, and trying to cleat it by working through the F-holes would be a pain, and quite expensive relative to the value of the instrument....

It may differ slightly on your Mandolin but generally this is the best placement I've found for my Weber. On my flat tops it doesn't seem to matter as much. Test out a number of spots, eventually...