The lodge is generally in the middle of the flooded area after they dam it or on the shore in a burrow with the entrance under water deep enough that ice won't freeze if shut if there's not enough water. They don't tend to live in the dam itself. They use all the water to protect themselves.

It should be noted that, aside from petty human concerns like flooded roads and properties, beavers are much better at controlling flooding and generally protecting ecologies than humans.

Saw a thing once, maybe Mythbusters, that showed how beavers are attracted to the sound of running water and want to stop it. In the show, they got the beaver to dam a speaker by playing river/stream noises through it.