In the web Inspector, when selecting something on your Artboard, it will pick the layer that is directly under your cursor. Depending on the structure of the layers in your document, you might find it difficult to select the specific layer you want.
You can specifically indicate which layer you want it to select by control-clicking on it. When doing this, you will see a contextual menu with all the different layers available at that location in the document. Simply click on the layer you want, and you will be able to see its properties in the web Inspector.
If you want to download assets they need to be set as “exportable” in the Mac app. They will be appear with a icon next to their layer name. When one of these layers is selected, you’ll be able to see the downloadable assets in the web Inspector.