Apple’s enterprise distribution and management for iOS apps
Each client will need to have an enterprise account. If they want to, they can share their distribution key with you, and you can build an app for distribution within their organization. If the number of clients isn't large, this shouldn't be a big deal -- you can just set up a different target for each client, and then use an aggregate target to build them all at the same time.

Over the air distribution is definitely possible. One way is by making the app available for download on a restricted access web server; another is to install it directly on any managed devices using mobile device management.

I believe most of your questions are answered in the enterprise program agreement. Give that a careful reading if you haven't already. Questions about how to interpret a legal agreement are probably off topic for this forum, but I think you'll find the agreement to be fairly easy to understand as legal documents go.

Update: Another option that's been available for a while now (though possibly not when I wrote the answer above) is custom B2B distribution. You can offer your apps for sale through Apple's Volume Purchase Program directly to your clients. That is, your clients would purchase through the Volume Purchase Program (essentially the App Store, but multiple licenses) and then distribute your app via Mobile Device Management or redemption codes. The down side of this plan is that Apple would take the usual App Store cut of the selling price.