Download games through internet, and then transfer to console?

you can Setup your 360 to download during power saving state, it will completly shut down once download is finished

Settings->system->console Settings->Startup and shutdown-> Background Downloads (enable if you like)

No. There is no way to directly download games from the website. You can only queue a download to start the next time you turn on your console.

Are you doing a network transfer? If so both console will have to be on while the transfer goes through. Also something that happens is the games will try to update before the transfer takes place and that will stop your install. I ended up just re downloading
most everything when I switched mine. Just made it easier. Lots of people have had this issue when doing transfers. My internet was fast enough that it wasnt any faster to do a network transfer. Plus it only transfers the core of the game and will still have
to download the updates. Hence where the issue of it trying to download updates before the game is installed.

You can redeem codes online at Xbox.com, but because the game is for Xbox 360, you will need to connect the console to the internet to download it. It's not possible download and transfer it from PC to USB drive and then to console.

Glad you managed to get it sorted.

Just to clarify for others with a similar issue.

When you transfer Xbox consoles i.e. upgrading or through fault etc. you will need to transfer over your licences for all content in order to play/use as normal.

Downloads, content and game saves can be played if you are logged in (on line) without transfer; however in the case of online apps as above (you tube, internet explorer) you will need to transfer your licence and in most cases delete the content and re-download
when signed in to the console using your profile.

Licence transfer as detailed here:
support.xbox.com/.../download-content