I understand that NPN refers to the switching being on the N-channel, or ground side of the circuit, as opposed to P-channel. Does this mean that both sides (input and output) are controlled-by/control the black wires (as opposed to the red)? I understand that this could work on the output, since one could have a given number of switching grounds and a common 12/24/whatever volt power bus, but how would all-ground inputs work? Don't all controllers switch 3.3/5V pins, not ground pins? Is this product description (both sides, input and output, being N-side) just the result of poor English or is there something that I am not understanding?
There are thousands of tutorials covering using the logic level MOSFET with Arduino. Do a search for "arduino logic level MOSFET tutorial". Learn the basics then move up to the more complex.
That means nothing at all
It might indicate open collector interface, but that's for BJT and not for MOSFET.
The input is an opto coupler that may be driven active low. High voltage level is described as 12-36V, most probably not related to the input pins. Or it means that an additional NPN transistor is required to drive the opto coupler.
Either you are familiar with electronic forensics and can find out the circuit diagram of the board and the input voltage, or you choose a better documented board.
That is the just the Amazon link. Those larger flatter chips in the lower-middle row are MOSFET transistors, and other listings/clones on ICStation, Ali Express, eBay, etc mention this.
I suspect they might be using "NPN" to mean Open Collector or Open Drain. It is a kind of digital output that only drives LOW. The implication is that the "NPN Input" has a pull-up resistor and will be read as HIGH when not connected and that the "NPN Output" requires a pull-up resistor or the output will be left floating when not LOW.