Regarding the workshop compressor it wont be filtered to standards for 'breathable' air and will contaminate the cylinder
Sites won't normally mention anything about you taking away air, however the air system may not be running after the last game goes out.
If you just take a cylinder to fill and take home then the site may not fancy it
If you take air home then you ought to put up a sign in your vehicle.
If its not a business vehicle then its not a legal requirement, but it's good practice and if something happens then the fire brigade will be grateful to know what you are carrying
(dont put on a permanent sticker unless always carrying compressed air)