This exists/existed in English too - you sit (yourself) and set something else, but that distinction is in the process of disappearing. It is more persistent with "lie" and "lay" and it's completely obsolete in the standard language for "stand" (which had "stell"). Usually it's easier for us just to "put" things :)
I don't mean over the course of the next few years, I mean over recent and future centuries. It is not common, at least where I live, to use the word "set" in this context. We would say "put" or "place", whereas in the standard language "lay" and "lie" are still commonly used as a pair, e.g. lay a blanket on the grass vs the blanket is lying on the grass. However, non-standard use of "lay" for the second example is common. You've kind of proved what I was trying to get at with your example though - "sit yourself down" and "set yourself down" are used without distinction, but sit is certainly more common where I am.
I'm curious how things are expressed and what they mean to everyone.
To me "puts up a bookcase" is more like putting the (already assembled) bookcase up against the wall and maybe slotting in a couple of shelves. Whereas "assembling a bookcase" means having to attach parts by using screws, lots of making sure that pieces are lining up correctly and possible use of one of those little hex keys.
What does "put up" mean where you are?