Login on xbox.com with your MS account and it should make you choose your unique gamertag, then you should be done.
You can have different billing methods for different regions for X1, but not for X360 as purchases from the console are often region or even IP locked. You can change your region once every 3 months, but you may lose some licences, it'll completely migrate your account and you're not guaranteed you'll be able to purchase those games.
On X1 it's much easier, you just have to change the region in bottom left (I think?) and get appropriate gift cards. You can't use your Microsoft credit balance to purchase 360 games, unless you buy them through a proper 360 console. Or, if you want to get 360 DLCs, you could launch the game from X1 and buy it from the emulated 360 OS, but that's just for DLCs and rarely for games (for example, BFBC and BF1943 can be purchased from BFBC2 menu).
My tip is: stay in your own region for 360 purchases, check if it's that worth it for X1/W10 ones.