Hi Jaromir, yes there's a lot of stuff, and bits of it are only relevant to certain designs, can get confusing!
The polarshield is an add on for an arduino mega. It uses pololu motor drivers, and a screen and sd card.
The adafruit motorshield can be used on an uno or a mega.
The instructable is probably the most "complete" guide, so that is building a machine based on an arduino uno, with an adafruit motorshield. I use beaded cord in my machines because I use a hanging counterweight so that the motors do not take the full weight of the gondola at all times, and that needs the sprockets to have a good "grip" on the cord - because the contact area is small. The same reason that der kritzler (and harvey moon's machine) uses toothed belts.
If you use regular bobbins instead of sprockets on your motor, you can use thin thread and just wind it up onto the bobbin. Pretty much every other hanging-v plotter uses this system. It certainly makes for an easier and cheaper build, because there are no unusual parts. The drawback is you can never turn it off without the gondola coming crashing down, and setup is a bit more awkward - the polargraph software doesn't have a "fast wind in" or jog feature. You could have a look at dan royer (marginallyclever.com)'s system for that - that's the model he uses on his machines.
Servo pen lift is necessary (well, sensible) if you are doing any vector plotting, but the machine will work perfectly fine without it - it just won't lift the pen.
The polargraph controller doesn't need to have anything connected. But once you've loaded your image, you need to use "select area" to choose the area that you want to convert to pixels, that will give you your preview.
Sometimes the software _does_ go mad, yes, I think it's a feature of controlp5 (which is the gui library I'm using) that it allows you to drag and drop the buttons around. It happens to me too. I can usually use the button correctly by holding down ctrl, or shift, or alt while clicking, but it never really gets back to normal. I have never figured out what triggers it.
good luck!
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