Roos

Deep shade, then water, and put the water in the shade so the two are the same job. A pool in full sun gets warm by noon and they will stand next to it rather than in it.

Normal, and the wings held out is part of the same system. They do not sweat, so an open bill and a bit of air under the wings is most of what they have.

Fix the pool, though, and not only for heat. Geese dunk their heads to cool off and to keep their nostrils clean, so they want water deep enough to get the whole head under rather than a trough they can only drink from. A cheap paddling pool in the shade is the single best thing you can put in a goose run.