You can retrieve ADRs here: http://disneyworld.disney.go.com/store/retrieveReservation
I've always gotten an e-mail listing my dining reservations after I've checked out, and it lists each ADR with the corresponding confirmation number.
I believe you only get that if you enter your e-mail address in at the end (or maybe the beginning) -- there's a box you have to actively select to get that e-mail confirmation, from what I recall.
I've always gotten that e-mail automatically after I fill out my info (name, phone number while on vacation, e-mail address) for each cart that I check out. There was never a box to actively choose to get an e-mail confirmation.
I put them in the cart and checked out. I never put my confirmation in, but I had been able to access them each time I went to the website. Now they're gone though.