Here in Canada a lot of our energy is hydroelectric and I know there's a lot of talk about this sort of solution in Quebec.
In this situation, you'd simply never let the water flow through in the first place. This amounts to releasing water while generating energy, only to pump it back. The net result is just a lot of wasted energy.
Pumped hydro requires two reservoirs. There's no way around it.
For that, you need water that can be pumped back upstream, that is, you need also a lower reservoir. If your dam discharges directly into a river, you won't have enough water downstream of the dam to pump back upstream. Not all dams have suitable geology to form another reservoir below them. An alternative would be to create a new reservoir higher than the dam, using the existing dam as the lower reservoir, with a set of tunnels with the reversible generator/pumps linking both reservoirs (this new upper reservoir does not need to be in the path of the river).
Yes. You need a mountain lake and a low level lake close to each other, and the lakes aren't useful for much else because the water level goes way up and down every day. The US has 10 pumped storage plants bigger than a gigawatt, and few attractive sites for more.