Toilets, Sinks, Showers, Tubs and Disposals - Where to drain???

I'm installing a dishwasher. I have a question about where the dishwasher drain should be going.

The old washer went to the drain from the kitchen sink right above the p-trap. There is a little copper pipe that sprouts out from the main copper pipe from the sink above. This side of the sink is closest to the dishwasher.
The new dishwasher says that the drain should go into the garbage disposer. I see the where it should connect to the disposer and it will reach it.

Should keep using the current drain or use the other side with the garbage disposer? If I use the garbage disposer side, what do I do with this small open copper pipe? Does it matter?

Thanks

You are suppose to run the drain to an Air Gap so that the dishwasher is not directly connected to the waste system in your home.


You will have to cap the tapped tailpiece or remove it if you use the disposal.

Two more questions. The tailpiece coming off the drain pipe. To remove it, would I just have to replace the drain piece coming from the kitchen since its a part of it? Or what would be the proper way to cap it?