Applescript to drop down Script Editor in Menu Bar
The AppleScript code you're looking for is:

tell application "System Events" to ¬
    click (every menu bar item of menu bar 1 ¬
        of application process "SystemUIServer" whose description is "AppleScript")


As to assigning F15 to that, well you'll probably need a third-party utility to do that, as I do not believe an Automator service supports using just a single function key, (or any function key), as its keyboard shortcut.

I use a utility called FastScripts for many of my scripts and it will support a single function key.



Note: I am not affiliated with Red Sweater Software, the developer of FastScripts, just a satisfied user.