I am wondering if I will have drainage problems with this setup:
I have a double kitchen sink with two tailpieces that connect to a horizontal piece and a p-trap going from there to the trap adapter at the wall. The horizontal piece has the typical dishwasher inlet on it. The trap adapter, where it stubs out from the wall on 1.5" ABS, is an inch above the horizontal piece. I am assuming the horizontal piece and the trap would always be full of water, which is probably not ideal, and also the DW inlet would always be submerged. I don't actually have this installed right now, but it is what I am looking to install. No disposal.

 

You will not have a problem. The trap is always full of water. This is so that sewer gases do not enter through the sink.

 

you will have issues imo.. lower the arm in wall

 

"horizontal piece has the typical dishwasher inlet"
As far as I know this is not typical.
Typical is that the dishwasher inlet is vertical and part of a sink's tail piece.
This directs the water down into the drain so it does not flow up into the sink.
It does not look like you have room to do this.

I would connect the dishwasher drain into the 1.5 inch ABS drain.
Just make sure that the dishwasher drain hose is placed so that some water stays in it re: this will act like a trap.
Most units end up being like this as the hose sits on the floor and then go up to the bottom of the counter top or close and then down in to the drain,

Then I would hook the trap up between the sinks and into the ABS drain.

Probably more than one trip to HD etc. for fittings.

Lowering the ABS drain would be a better solution but sometimes this is just too much work or not possible without ripping apart and redoing major plumbing alterations.

 

The ABS arm in the wall is definitely too high and must be lowered.