You cannot download Xbox 360 Arcade games directly to an Elite console without using an actual Xbox 360. The marketplace ties purchases to a specific console ID and profile at the moment of download, writing encrypted files that only the Xbox 360 OS can authenticate and store. No PC tool can replicate this process for offline transfer.
Games are region-locked (NTSC-U, NTSC-J, PAL). Your Elite must match the game's region or be modded to be region-free.
Bottom line: You need one Xbox 360 console (any model) to initiate downloads. Then USB/HDD transfer + license transfer = play on your Elite.