Buying Digital Games using a Credit Card of a different region

No. The payment method would have to match the region.

I don't think you can... Last time I did it, it just screwed up my billing address and it doesn't accept any CC from my region. I'm using Gift Cards by now. If you still didn't change your Xbox live region, then I suggest you keep it this way.

If your account region and actual region doesnt match you will continue to have problems downloading content especially triple a titles

If you're trying to purchase the free games on the website, then that's why it's asking for a payment option. The website hasn't been updated to work with the new currency system yet so it requires a payment option. If you make the purchase on the console,
you don't need to add a payment option.

I use gift cards too, but the problrm is lately, you have to enter ur cc info, for example for the games with gold out now, you have to enter a payment option and the only one is credit card, even though games with gold are free. About 1 month ago xbox converted
all my microsoft poimts into dollars and I can't even use them without entering my credit card info, I guess this has something to do with the whole "no valid payment options" error