The Wedge is merely a visual indicator for the Active Layer. Nothing is selected. The wedge is not an indicator of any artwork selection. It is no different than the highlight color when a layer is active. It's a remnant from when the app didn't have a color UI.
The Meatball directly indicates artwork selection.
A double-ring Meatball indicates that ...
To be able to cut you need to make sure:
layer is not locked
layer is not smart object. generally, needs to be a raster layer (no vector shapes)
you need to have something selected (i don't see anything selected in your image)
you can not cut from a Smart Object layer, you can copy, just can not cut. Have to rasterize it to cut.
Similar problem: If you can not cut AS WELL AS copy a selection, you must be working on a shape, and still, you got to rasterize it.
Yes there is setting you could use, but not with the Brush tool. Select the Clone tool instead, then in the Tool Options choose a brush shape, set the Source option to "Pattern", and Alignment to "Aligned", and choose a pattern to paint with from the Patterns dialog.
There's also another technique. You could fill a layer with a pattern, or even just open ...