Well your treasury carries overSince Cities become towns at age transition one could argue gold doesn’t fully carry over as well (barring Economic
Legacy golden age).
They add Tiles, but do they change the Population of the city?They do, but I’m now not sure if it’s all new buildings (overbuilding definitely doesn’t count) or just a new Urban District derived from constructing a building. I suspect the latter, so it’s fair to say a portion of building Production is Food.
Having trouble finding definitive proof one way or another due to the woefully low-information City screen, but just compare any city or town’s pop with the number of Improvements. It’s particularly clear in the latter portion of the antiquity stream before specialists come into play.
[although even if they do, that population does nothing...its the building that does something]
Production and Food are definitely the two Separate ways to increase an individual city's output.