You need a vent for EACH device or you will siphon the other traps when you run the WM. You may also find that code would require you to make the drain line larger if you add other things to it. There's a limit, based on the drain line diameter, how far the trap can be from the vent. And, there are some pretty specific rules about where the vent and the rest of the drains can be installed. BOttom line, I don't think you'd find any inspector that would pass what you're interested in doing with your proposed plan, and for good reason. You need a separate trap on the WM drain, and it must also be vented. That doesn't mean you must run a new vent all the way through the roof, but there are some specific rules on how to join into existing ones, too.