A lot of people get venting wrong. A plumber should have got it right, but you never know. No place that I know of allows wet venting between different levels (basically what you have). Could it be that the WM and tub were added after the house was built? I know that often in basements and such when plumbing is added, either:
1. venting is too hard, so they don't do it
2. they think that connecting into the stack within some distance is good enough to vent it.
In my house (built in '68), nothing in the basement bathroom or laundry was vented. I had similar issues. In my case, the trap in the shower would get the water pulled out of it and you would get sewer gases coming in. I had to cut concrete, re-work the plumbing, and run a vent from the basement to the attic and out the roof (which was a pain with 2 finished levels above the basement). Now, everything is good.
I'm not a plumber, but I would do the wet before the backflow preventer. In fact, I would probably remove the backflow preventer as it probably isn't doing anything for you. If you had a backup, it would just overfill the laundry tub. The problem with backflow preventers is they are mechanical and anything that may cause them to stick wll cause them not to function. It may not be as much of an issue on the WM or laundry tub line as a main line, but I suspect lint and such could cause it to stick. If you keep the backflow preventer, I would also add another on the laundry tub side.
The kitchen waste would come in "below" the lowest fixture, not anywhere above.
Vents go up, and waste goes down. A kitchen sink can't use a downstairs vent for a drain.