As of now, the log window is always shown with the script command called if a script is invoked from Tools / Integration. It would be nice if the window was not shown if the script itself doesn't produce any output in the log.

Just uncheck "Auto activate" in the log popup.
Additionally, you may want adjust Filters to hide unnecessary groups of messages.

Ah, I hadn't noticed those settings. But I do want the log window to pop up for relevant messages. I just tried to uncheck Script in the filters menu while keeping Auto activate. This makes the log window still pop up, but no new message is shown.

Wouldn't it be a good idea to have Auto popup look at the filter settings and only pop up the window if the output is shown?

I agree.
It will be done in 4516