Given the resources and experience available to Google, the obviously high demand for a product such as this on launch day, and the absurd number of errors and issues at almost every step along the way, it's hard not to consider the possibility that Google is doing this on purpose. I mean really -- a caching issue that causes the phone to show as sold out every so many refreshes? Then again, all they'd really be accomplishing is making the "Play Store" look dumb and unusable.
Anyway, I digress. Here are some tips to getting through:
1) If you haven't ordered something via Google Wallet before, go here (https://wallet.google.com/manage) to fill out your payment info and billing address in advance. It's much faster and more reliable to do it here than to try to do it in your shopping cart while buying the phone.
2) Once the phone is in your shopping cart, just repeatedly click "Proceed" every second or two, even if the button is gray. Eventually you will get a popup that will let you order.
3) It's normal for the phone to have disappeared from your shopping cart if. Just go back and add it again. It's also normal for the phone to show as sold out. Just refresh until it says you can buy it again.