How can I fill the areas marked in the diagram with a fill or gradient? (The lines shown in red are Objects, not Strokes).
As long as the objects/lines form enclosed shapes (and have no gaps), you could use the bucket fill tool. If there are open areas/areas with gaps that you want to fill, you could just draw some lines to close them with the Pencil or the Bézier tool.
The Bucket fill tool options has a Close Gaps function, but sometimes it doesn't work so well. For the best results ensure that all areas you want to fill are closed.