Sounds like you may be confusing an air gap (that installs above the counter - well, actually through the counter) and an AAV. If you did use an AAV, it would be as high as possible under the counter where you could still unscrew it to replace. There a lots of diagrams of other projects where this has been done to go by.
Yes, that does look like an s-trap. Even if it had a loop vent (which would pass code most places), it's still not installed properly.
The disposal will have a DW inlet port, you can use that. You need a baffle T on the part connecting the two sinks to prevent the outlet of one from flowing to the other sink. The disposal acts like a pump, and without that, you'll have problems.