The question in the title is easy to answer - most large galaxy instances run (most) jobs on a connected cluster which is effectively 'background' I guess so "yes - all of them on cluster backed Galaxy instances" might be a reasonable response.
The questions in the text are more complex, but (eg) AFAIK there's no technical reason why some batch process owned by the Galaxy user shouldn't issue a
sh run.sh --stop-daemon
at any time to stop Galaxy. Exactly when that might be a good idea is not so clear to me so I'm probably missing something here - the Galaxy process doesn't consume a lot of resources and is typically left running permanently so users can take advantage of the persistence and reproducibility. Tools are AFAIK not generally written to launch their own background processes because they are managed by the Galaxy job runner which interfaces with (eg) SGE or Torque to run each separate tool invocation and clean up when it has completed.