There's a thread below about plants that love wet soil. Along with their suggestions is mint (but it willgrow EVERYWHERE if you let it), Joe pye weed, butterfly weed, milk weed, cattails, pussy willows, grapes, golden rod, asters and willow trees (although some people consider willows giant weeds.)

Sorry I can't help you with the grasses. Personally, I can't see planting it, when I'm always trying to weed it out of the vegetable garden! LOL!

The plant with huge leaves is probably elephant ears.

Here is a link that might be useful: Wet soil lovin' plants

Butterfly weed would not do well in wet conditions. It prefers dry, sandy soil. Swamp milkweed (Asclepias incarnata) does well in moist to wet soil. Here is a short list of plants that do well in sun and moist to wet soil:

Queen of the Prairie
Buttonbush
New England Aster (can be aggressive)
Obedient plant
Sneezeweed
Joe Pye Weed
Marsh Marigold
Iris versicolor or Iris virginica
Monarda didyma
Sweet flag
Carex stricta (tussock sedge)
Turtlehead (Chelone glabra)
Royal Fern
Cinnamon Fern
Sensitive fern
Red Osier Dogwood
Boneset
Cardinal Flower
Blue lobelia