Could down load a Mod for a game onto a Flash Drive and then install it on your XBOX360?

No, you could download a nodded saved file though, or you could just mod your saved file, But to download a mod not in a saved file will not work unless your Xbox is Jtagged. So say you want to mod Black ops 2 online like everyone else has, It wont work unless your xbox is Jtagged, But if you want max arrows,Gold,Level,health e.t.c for skyrim and you have either downloaded someone save which has this(Modio) or if you actually modded your own saved file to do this(Horizon) then yes it would work.

You can't download a mod onto your xbox. What you do is just transfer your save via USB onto your computer and extract it through Modio/Horizon. You do all the save edits on your PC and then just replace the old save, while rehashing to allow your xbox to recognize the altered save. Then you just plug it back in and but it back onto your hardrive. That is the only way to do it on consoles, but PC has way more potential, espeically on Nexus where they mod the living crap out of Skyrim. You can't do anything like go into the coalesced bin like computer users. I just use the save edits for Mass Effect (including hexing), but it works for every game.

I do it all the time

Really easy