How can I hook up my dishwasher drain properly?
I have had my Bosch dishwasher for about a year using the same setup you contemplate, so I'm familiar with the exact drain connector you have on the dishwasher hose.  On mine, there was something of a taper between the 1/2 and 3/4 section and being rather flexible rubber, and I had no difficulty stretching the 1/2 section of the fitting to fit on the 5/8 gap fitting.  I then tightened down the hose clamp with a hex bit on my cordless driver.  The clamp was tight enough to somewhat compress a groove into the rubber, and I tugged on the thing as hard as I could.  It isn't coming off IMHO.  I'm at work, but can send a picture later if you need.

I understand you are worried somewhat about the quality of the connection.  You could get a 1/2 to 5/8 adapter like this:



But that would entail 3 hose clamps and a section of 5/8 ID hose to connect between the adapter and air gap.  That seems like more points of failure.  At any rate, I think you already have what you need to get this working.