The lav can wet vent the toilet, using a 3x2 wye or combo fitting, running the lav in 2", and then terminate with a 2" AAV with access.
No need for a 3" vent stack.
The washer upstairs on the separate 2" going upstairs can only have an AAV if there is at least one open air vent going through the roof. Though I would check on that, because there might be positive pressure on the washer vent when it pumps out. It's best go go atmospheric there.