Need advice on trapping/draining/venting a replacement washtub & standpipe

I want to replace my existing concrete double basin washtub with a new single plastic unit. At the same time I also want to add a standpipe for the washer to drain into and also drain the softener into something other than the washtub. Right now my washer and softener drain into the washtub.

My question is draining/trapping or how to I guess. As drawn in the picture, can I add a standpipe to the left of the washtub, elbow over (no trap) towards towards the washtub, go into a wye which is also connected to the washtub drain, into a trap, and then into the drain stack?

Is there any reason both the standpipe and washtub drain need individual traps?

Any particular recommendation on what to do with the softener drain?

Do I need to do anything with venting?

I can accomplish the work no problem, I just need direction.

Any recommendations appreciated.

Picture two shows the drain piping for a shower in the bathroom just above. Is the white gizmo at the top an AAV? Continuing to the right is a long run to the kitchen sink drain.

Thanks!

You are on the right track.

Make sure you use 2" drain line all along the horizontal run and the stand pipe for the washer. The new washing machine pumps move a lot of water in a hurry and time after time I see 1-1/2 drain pipe back up water. Even though the water should back up into your tub...better safe than sorry.

Keep that trap within 30" of the stand pipe, at least per the code I have to use.

Yes, in pic 2 it looks like an AAV.

Good luck.