The dishwasher drains into the disposal. I would be tying the clothes washer below the kitchen sink/dishwasher inlet and above the bathroom inlet.
I have heard that I may need to separate the sink and dishwasher drains, creating a trap arm for the dishwasher
The dishwasher needs to go up, to an air break, before it goes down to the garbage grinder. I like the deck mount air breaks, because you can see if there is a problem. It also keeps ground garbage from back feeding into the dishwasher.
Vertical wet venting is all about fixture units and UPC (908.1) doesn't mention bathroom groups.
Bathroom groups come up in horizontal wet venting in the UPC and in the IPC under vertical wet venting, but I am not schooled in IPC and have no interest in pondering the differences. https://codes.iccsafe.org/content/IPC2018/chapter-9-vents