Brain did a great job of addressing your concerns, let me take a shot at it.
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I am concerned that the dishwasher drain hose wasn't installed correctly
Your description describes a typical counter top air gap installation (see image. Because they can fail we use a high loop, (see image) in my area.
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the disposal doesn't have a p-trap connected to it.
Disposals don't have a built in trap. ( here a picture would be helpful) However there should be a trap before the pipe enters the wall. Is it there?
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the dishwasher never really drains out all the water
It won't! Once them pump stops all the water up to the air gap will drain back into the sump,
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the dishes don't get as clean as I think they should
Is this the water coming in? Describe what it looks like and what you're concerned about.
Back to you, Tom