You should be fine using a standard multimeter but you may want to remove the battery so that there is less chance of accidentally shorting a power rail to something that can’t handle that. If your multimeter has a continuity test mode you might be able to just check continuity between each pin of the switch and the flip the switch and check again. Without a continuity mode, you can instead use a resistance mode and check the resistance between the pins. If the pins are not connected at all, the multimeter will likely show something that looks like OL (for overload) but if there is a path between those two pins somewhere else on the board, the multimeter will show the resistance of that path. If the pins are connected, the resistance should be very low (potentially less than one Ohm). On my Chestnut L5 the middle pin is connected to the upper pin when the switch is in the upper position and to the lower pin when the switch is in the lower position. Based on your description, it seems that the middle pin on your switch may always be connected to the upper pin.
Be very careful though, even with the battery removed, there may still be capacitors on the board that hold a charge and could be shorted to something causing damage.
That loose pin in your phone is a bad sign though I am not sure where it might have come from.