Many thanks for the helpful summary!
I was at the Arequipa branch today (no appointments in Lima, even though there are apparently people that can facilitate appointments - but it's also a nice excuse to visit Arequipa, very nice city!) and everything worked out as planned.
As previously mentioned, they required both a copy of my national ID and the declaration from which the Renta (for the Rentista Visa) derived, and even asked questions about it.
An utility bill was not required, but it states at the entrance that this can be requested in case of doubt about the address.