Two consoles, one gold subscription

The Xbox One can share a gold subscription, but only on the same console. If they're playing on two different consoles with two different accounts, you would need a separate gold subscription.

Home Xbox also includes game licenses. If you own the game digitally through the Xbox Store, you can install it on your Home Xbox and on any other console you sign into. So, in the case of "Destiny", if you bought it on your console, you would have two
licenses - one for your Home Xbox and one that travels with you to whatever console you sign onto.

This does NOT work for games you buy on disc, since the disc is the license to run the game.

As a follow up I assume the other console im signing in to will require another copy of the game. E.g If I have bought Destiny on my account though my Home xbox i will need to buy another copy for the other xbox one console to play together or can I redownload
to the new console.

Yes. Your girlfriend will access your Gold subscription through the Home Xbox feature, while you'll have access remotely through your Microsoft account and your console's access to the Internet to verify that subscription.

Unfortunately, the Microsoft Services Agreement prohibits sharing of account details with others. (Technically, Netflix does too … just sayin'.)

as long as she is on the home Xbox and you are on your profile on a diff console you should be fine