I can buy games and download the live another day?

No worries. You can certainly purchase the items and then download them later. When you get your hard drive simply go to Settings>Accounts>Download History on the console to start the download

The big issue is that I got my xbox from the UAE and therefore Xbox Live but I am living in Pakistan where original xbox 360 games are almost non-existent. Either I will have to wait another month for it, or I can buy and download it earlier but I think
I'll wait for another month. Thanks for the help guys!

My Brother doesn't own the main Game, just the DLC which we bought because it wasn't working with my account logged into the store.

But it seems to detect the ownership if I can quick travel to it and it prompts me to install instead of buy right?

Another issue is that the town where my parents live has terrible Internet. I'd need 2 to 3 days to download the 80GB again and I'd like to avoid that if possible

Is it fully installed?! Also, there's a "day one" update every time you buy a new game MAYBE it's still downloading the update. Another thing that MIGHT help is try to delete the game & download again.... Sounds terrible but it might help.

So if I buy another live account(GT), play with the live account(GT), on original console game was downloaded to it will work just fine.. Then i take a copy of that game on a usb and play on different console with original GT enabling two consoles to
play same game?