As per the summary, Yakuake currently only allows retracting/down-droping the application with F12. I think it would be nice if F12 would also be used for initializing Yakuake if it is closed.
Technically speaking this should be doable. UX-wise I'm not sure. I've found myself wanting this too, but I can also see how it could be unwanted that a not running application is started
KRunner also does start when pressing the shortcut and not running already