It is easy, as of March 2016:

As of the March 2016 update, the Xbox store has an "Xbox 360" section, allowing you to access backwards compatible Xbox 360 games directly from your Xbox One. Keep in mind that the Xbox One can only access Xbox 360 games that are backwards compatible.

Something to note, concerning Games for Gold:

If you have Xbox Gold, and wish to take full advantage of the "Games for Gold" freebies, it is worth considering that any game that is not yet supported by the Xbox One will not be accessible through this means. You still have other options to 'purchase' the non-compatible games, while they are free, in anticipation of the games later becoming supported.

If you have an Xbox 360:

Too easy. Simply log in to the store from your Xbox 360, and make the purchases from there. From an Xbox 360 you will have access to the entire Xbox 360 library, available to your region. You can also be signed in to an Xbox 360 at the same time as your Xbox One, so you can run it through your Xbox Ones 'HDMI in', to make life easier. If you are borrowing a console for this purpose, you can opt to require password at each sign in, if you have worries about illegitimate access to your account.

If you have no way to access the store from an Xbox 360, and the game is not yet available for backwards compatibility:

Log in to the Xbox Store from a computer, sign in to your email account, and purchase the game from there. Ensure that you are accessing the right Xbox store for your location. The above link is the English store, and other stores may vary in content and compatibility.

I'll try to answer or address all your points.

First off, you can buy 360 games on the One. The reason you aren't able to is most likely because of your payment method. I personally think the easiest way is to buy a currency card and redeem it.

Once you own a content, no matter where you bought it, you can download it to any console your profile is logged in on. Should the title have been backwards compatible, it will also show in your Ready to Install list on the Xbox One games app.