The search bar in the appilcation menu is meant to be used as QuickLaunch (awesome little application that should be installed as defaut IMHO) : you type in the name of an installed application to start it. The second one in the main Dekbar is meant to launch the query. But you’re quite right, the optimal solution is to have only one search box (the main one) able to launch installed apps AND perform a query.
I am not sure to understand what you mean ? The main idea is to «extract» what’s inside the blue leaf menu to make it more discoverable (as some reviewers didn’t find it).
I totally agree ! I absolutely don’t want to loose the Haiku GUI feeling, and if that’s the impression you got then my mockups need more work.