Must all firewood sheds have airflow?
Firewood needs to dry out. It won't dry out in a sealed shed nearly as fast and completely as it would in an open-air setup. 

Worry about moisture in the air 'getting back in' to the wood isn't really much of a concern. It's not atmospheric moisture but the moisture that's in the wood initially that's the concern. 

If you completely season your wood outside for a year or so, THEN bring it in to the airtight shed, that could work, but seems redundant to me.

If the only real concern is rats, then I think building a shed is equally problematic as they will find comfy homes under the shed or even inside after they chew a few holes in it. 

The reason most firewood isn't stored in a shed is that it's just a waste of a perfectly good shed. :)