Using Filter>Distorts>Polar coordinates
I assume the real goal is to map the image to a sector, and that the path is not really necessary.
If we apply the Polar coordinates filter to a square image:
We get this:
The final image is the same size as the initial image
The whole image is wrapped on a full circle (360°)
The height of the image is mapped on a ...
As far as I know, in GIMP this would require some manual work. There's no automatic arc distort tool. So if you're up for that, it's entirely possible. Unfortunately the Curve Bend filter in GIMP is difficult to use if you want some accuracy.
In GIMP use the shape you made as a guide. Set this image as the top layer, and set the blending mode to "Multiply" ...
If the things to move are each on their own layer (use the selection to copy/paste them to a new layer if not), then you can chain-link the layers, and move one of them, which will make the other ones move the same way.
Gimp likely is not aware of the capabilities of the receiving end of the copy/paste so it copies what it has.
Easy work around:
Add a layer below your image
Fill with the default background you need
Both answers use the ofn-path-edits plugin:
Answering the title of your question:
Starting with this:
Control-Shift-click on the two segments at the ends of the group you want to remove, to delete them and isolate the group as a "stroke".
Make a rough free-hand selection arounds one of the ends of the stroke
Right-click the path in the Paths list, and ...
There is no way to flip non-parametric brushes.
The brush rotate options are only available on the "parametric" brushes (those that you can create with the brush editor). Since all these brushes have a symmetry you can achieve a flip by the suitable rotation.
There is no way to set keyboard shortcut to rotate non-parametric brushes.
You can set keyboard ...
The channels become grayed out whenever the image is not using the usual RGB mode (in other the words image is grayscale or color-indexed). To change the color mode, click "Image" -> "Mode" -> "RGB". Then you will be able to use the Curves tool to its full extent.
Anything "Auto" is not going to be the same on all images. So
you can accept that images are going to be transformed differently. This can be a good thing in some cases.
you want to apply the very same changes to all: you have to use an explicit/manual tool on the others (Levels/Curves...) but if you use an "auto" tool on the first, Gimp won't ...
A quick workaround - bring focus to the first link of the first layer and select it --> down arrow on keyboard --> enter button --> down arrow --> enter --> arrow --> enter . . . . . I did 150 layers in 20 seconds or so . . . .