I'm not sure how this works but I installed the first of 7 files via GOG Galaxy. The Play button is grayed out. And the status says it's still allocating 0.0%. What should I do?

Uninstall GOG Galaxy, and reinstall the game manually. Once installed, first patch with the enabler file (patch 101), then the other patch (106).

Once all 7 parts and 2 patch files are downloaded, do I manually move them to the appropriate gaming folder which is the following after uninstalling the client; /GalaxyCient/Games/Witcher3/bin/x64 ?

And then just run the exe file?

No. If you have uninstalled Galaxy, those folders should not be there. Create a new folder for all of your downloads. Something like C:\Downloads\Games\, create sub-folders as necessary, then place all of the files in there. Then run the installer from your new folder.