https://store.sonyentertainmentnetwork.com/#!/en-us/home/games
They've finally release their new website which lets you purchase, select downloads, etc via any internet browser. Sadly, you can NOT tell your PS3 to turn on remotly via website yet.
Still, as anyone with a horrendously slow internet connection (ahem) can tell you, the new PS Store on the PS3 itself can take upwards of 5 minutes just to load everything. This option is significantly better/faster for people with slow internet.
Also, when you make purchases/select downloads on your account on that website, they will
EDIT: You can also redeem codes via that site too. Just tested that, worked fine.
EDIT 2: They do not appear there. They appear on the PS3 in Account Management -> Download List. Then you can click to download them. It still 100% avoids using the PS Store on the PS3, so it's still a FAAAAR better way of browsing PSN.