I have a ring but then I wanted one. It makes no difference to the official-ness of your marriage so if you'd rather not wear one, don't :)
A quick google about confetti brings up this - "The custom originated with the ancient Assyrians, Hebrews, and Egyptians as a symbol of fertility and good wishes for a bountiful life. Although nowadays, you?re more likely to throw confetti, flower petals or the ever-popular bubbles. That tradition began due to the fact that rice is bad for birds to eat."
I didn't get married in a church. It wasn't a registry office either though. We did have the wedding march played. You're allowed anything you want as long as it isn't religious and I always wanted to walk down the aisle to the wedding march. I know a couple who walked down the aisle to an orchestral version of Metallica's "Nothing Else Matters". Just like the ring - it's your wedding, your choice.