Just check the videos page. Make sure its showing the latest videos, and it's just oldest first (so starting with Wii U Launch: Wi-Fi Woes and Little Inferno). Also if you download them, they're numbered.
How do you prove to them you bought the game? Take a picture or something?
More or less. Yeah.
When I ordered Assassin's Creed: Revelations back in December (during a boxing day sale), I ended up with a Signature Edition of the Game that didn't have the code for unlocking extra content. So I took pictures of the disc and attached a copy of the invoice and e-mailed that to customer support and they sent me a code.
This is a much better resource than my list. Half of the reason that I haven't updated it in so long (around a year and a half ago, and I think all those download links are broken now) was because it started to get unwieldy and difficult to use. The other half is because I got real lazy. This is way easier to navigate.
Criterion's games are the only racing games I play any more, and this is awesome for both the studio and franchise. I just hope they'll be allowed to make a straight Burnout game again one day. I need those crashes.