Thought installing a new sink would be a relatively simple job. As I look to get started, the new sink did not come with the hole and equipment for the dishwasher drain. I think its called the air gap. Question, can I bypass the airgap and connect the hose directly from the dishwasher to the disposal?

You can as long as you have hose from the dishwasher loop up higher than the drain from the sink. This will keep water from going back to dishwasher