QGIS - processing area and calculating volume of polygon with know elevation from a DEM

There wouldn't be a specific tool, but a work flow you could follow to do this. I have done something similar in ArcMap, and can describe the concept.

Firstly decide what the fixed elevation is of your dam. Then you can convert your rectangular polygon to a raster file (matching parameters such as cell size with your DEM). There should be a tool for this.

Now perform a raster function where you subtract your watershed from itself so that you have an empty watershed raster (raster calculator aka map algebra). This will be used later to construct the water elevation raster file. (because: empty raster + 100 = water elevation raster at 100 feet)

note: when creating your empty raster make sure to clip it to whatever you think will actually be the reservoir

Now you can display your DEM, and the water elevation raster on the map. If you perform more raster calculations you can generate a new raster that shows the depth of the water. (water depth would be the difference between the water elevation raster and the dem)

To find area you may consider the basics. area= # of cells * length of cell * width of cell so you could use the water elevation cell count and cell size for this input. Most cells are squares so the length and width values would be the same.

to find volume you can use raster calculator and use the water depth raster as an input then multiply by the length and width of the cells.

You may need to play around with this a bit, but that would be a basic processing flow for your needs.