Having installed and brought a BTG Core wallet up to date on a USB3 hard drive, can I take that drive, connect it to a different PC and run the wallet from there?
This would be preferable to having to download the entire blockchain again on a fresh install on the different PC 
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Yes, you can copy the synced data files to another computer and they will work.
Be CAREFUL to shut down the wallet at the old & new locations, etc. If you pull a USB drive while a node is live, data corruption is likely, and then the node will need to re-sync.
If you have space for it, making a backup copy of the data isn’t a bad idea, either. If you have a corruption problem or drive failure, a backup copy will let you get back up and running much more quickly, even if you have to sync several months of “latest” data to get current.