To clarify, the registration is processed immediately at city hall. It just takes some time to make the photo ID card. The certificate of registered matters is basically a temporary thing to show that you're legally registered while you wait for the card to be made.

I had trouble using a certificate to get a phone; the DoCoMo shop in central Tokyo said they needed "a card." Others I know have reported success with the certificate. It seems that this depends on which employee you get and how much of a stickler for the rules they are. (As I noted on another thread, you can get around this by having a friend or host family member go to the phone store with you and apply for the phone in their name, but providing your credit card as the method of payment. The only catch here is that they must approve any later changes to the contract, e.g. cancellation.)

Banks are generally fine with the certificate so you shouldn't have much trouble in that department.