If you run the kitchen sink as Mr. David drew it in red in post #4, then there is no wet vent. The pipe below the arm to the lavatory drain is not a vent, it is a drain. Since the kitchen drain pipe ties into the drain, there is no wet vent. If you look at your original drawing, the lavatory arm is tied in, and the pipe above that is a vent, with the kitchen drain tied into it.
On the shower, run straight at the washer vent as you have it drawn. Before you get to the vent, add a wye into the drain line, and off of that wye turn another stack up in the shower wall. On that stack, catch the shower as Mr. David illustrated in red in post #6.