Set OBS to the size you want, and then fit the source to the canvas. (Right-click -> Transform -> bottom few options)
The cable itself is probably not the problem. Unless it's super long, it's probably passive. Just wires and nothing else.
Everything else (because they have active electronics in them) has the potential to be like I described above. Convenient dividers from convenient fixed clocks...
DisplayPort is its own thing, not tightly connected to the improvement path above, and so it needs a significant amount of logic to convert to and from something on that path. From one point on that path to another still takes some logic, but not as much. The special case of DVI -> VGA specifically, from a graphics card, can be a straight wire because that DVI connector actually outputs both formats simultaneously on different wires. That conversion logic is already in the graphics card, but not all graphics cards do that, as evidenced by the absence of those pins in the connector and so the adapter doesn't fit. HDMI doesn't have those pins at all...