Couldn't you configure the existing panel to do what you want? Add and remove widgets and resize it and you should have what you need.
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No: there is no krunner widget I can add to the panel and AFAIK no way to hide/show the panel via a keyboard shortcut
As far as I'm aware, no launcher (in the style of KRunner) exists which has all those features.
With some effort, you could implement a lot of this functionality using a "generic" launcher (like rofi, for example), but integrating a system tray will be close to impossible (unless you're willing to write a program which acts as a system tray and provides those launchers with enough information to display a useful representation of the tray, could be cool).
You can create a panel in edit mode and put whatever you need (clock, system tray, application menu) on it. Set size to fit content and make windows go below which results in a tiny floating panel for example.
AFAIK there is no krunner widget I can add to a panel... but regardless: can I have the panel show/hide via some keyboard shortcut?
(If they can't be together, I could live with alt+space => toggle krunner and, for example, alt+shift+space => toggle panel)
I'm not sure if you can show/hide like that, but as a workaround you can toggle auto-hiding with a qdbus command, and set a keyboard shortcut to run that.
GNOME
Edit: Sorry, I may have misunderstood the question as I'm not entirely sure what a launcher is in the context of Linux, but I've been using GNOME as my desktop environment for years and it does everything asked for here.
IIRC krunner works as a commandline tool, so maybe you could do what you want with something like yakuake, or even make a wrapper for krunner with those additional options.
It seems like something I could use as well. I'll note it down to take a look later.