Hi Rosalind,
Having worked with computer boards for the last 40 years, those kind of switches are quite common. They are used to configure various settings on the board easily without having to rewire the board or place/remove jumper connectors.
Since we don't know what your switches configure without a theory of operation document and/or a set of schematic diagrams, I'd say you just have to assume that the switches are supposed to be in the state you found them; it's not likely that whatever issue you're having with the circuit board is related to those switches since under normal circumstances there is no reason to change them.
On the picture of the circuit board you included we can see that the top four pins of the switch are all connected to the same power plane but the only way to even get a hint as to what the switches do is to follow the circuit board traces and see where they go, in particular switch number two since that's the only one turned on. Since we can't see any traces on the top side of the board there, that means those pins are being fed through the board to either the other side or possibly an inside layer. If it's on the other side you might be able to follow the circuit, but of course if it's on a layer between the top and bottom that makes it a lot harder to follow. Sometimes putting a bright light behind it will allow you to see the interior layer but it depends on what other circuits are around as to whether you'll be able to see them or not.
Anyway, like I said it's doubtful your problem comes from the switches; your answer b) is the correct one. In trying to fix a circuit board you really need reference materials like those I mentioned previously in order to figure out what parts are working and what parts aren't.
What is the specific model number of your sewing machine? I have no idea if technical information is available for it, but we would need to know the model in order to have a chance to find the right documents for you.
2 
63 
99 