The document appears to be in Outline Mode.
Outline Mode shows the spines of paths in order to more easily view construction. It doesn't show fills or color.
Switch to Preview Mode (View menu), if you wish to see fills and colors.
You may have inadvertently hit the shortcut for Outline Mode (Command/Ctrl+y)
It's probably best to create a vector layer. Then make a circle. Then select the circle with the Select Shapes tool.
Then you can change the stroke and fill in the Tool Options. Because the object is a vector, you can do this as many times as you like.
Although changing the fill here looks like I'm using the Fill Bucket tool, it's not the same thing. The ...
If you have a path that looks like this:
You can fill the area below it by defining the area you'd like to fill.
I would suggest using the pen tool, click on one of the end points to continue the path. (Notice the slash icon that appears below the pen cursor.)
You can hold shift to constrain the lines you're adding to right angles, and click to draw the ...
Inkscape: If it's a path you can use it to divide a rectangle to two parts. The function is Division in the Path menu. Use a copy because the path vanishes.
No guarantee because I cannot be sure if your curve is a path. It can be a group which contains many different parts. Division needs a single path and the rectangle or a closed path below to be ...