That depends entirely on the depth of the river and the depth of the reservoir. If the average depth of the reservoir is deeper than the average depth of a river there is less surface area.
Given that Earth averages towards smooth, the rate of width increase overwhelms the rate of depth increase when dams fill.
The only exceptions might be narrow canyon dams with nowhere to spread laterally, or dams on already-wide rivers. But those are rare.
No it doesn't.
The only thing that matters (in this oversimplified calculation which only takes into account surface area) is average depth of the freshwater while it is on land. If the reservoir is on average deeper than the rivers the freshwater otherwise would be flowing in, there will be less evaporation per liter of freshwater available for use.
Now a dam also increases the total amount of freshwater that's kept on the land in a steady state situation compared to if the water flowed free into the sea. It would be absurd to count this as "extra evaporation" when this extra freshwater otherwise would've simply be lost when it would flow into the sea instead of being kept in the reservoir.
Furthermore the fact that dams increase evaporation though this mechanism is an easily verifiable scientific fact. https://riverresourcehub.org/resources/how-dams-affect-water...