Do I need the host PC to be powered on?

I have the Shuttle 48TB Thunderbolt 3 here. There are long running background tasks running on the Shuttle - Media Patrol and Redundancy Check. Do these require the G Speed Software Utility to be up to monitor those tasks?

No they will run without the software being actively open.

Do I need the Windows laptop to be powered on then?

Yes, the system would need to be on in order for the device to be on.