I asked Matt our Designer tutorials guy to look into these questions with me.
The first ones a bit of a mystery to us - because you ca toggle scale with object on and off and it resets to show two different states. Whereas that doesn't seem to do the same thing with other default brushes so this might just be some sort of bug with that particular brush. But generally speaking Scale with object usually works really well for me and hasn't brought up any other issues in the past.
Matt had the following to say regarding your 2nd question.
So my understanding with the transform panel vs using the node tool to change the line is that the node tool is redrawing the line in a new place or extending it ‘officially’ but the transform panel is dragging it and resizing it, so that kind of makes sense that by extending the W in the transform panel you are technically effecting the overall characteristics of the shape, but using the node tool always seems to me more like you’re redrawing it. I guess thats hard for me to explain but personally I think if the stroke width increased by just dragging out the end with the node tool that wouldn't make as much sense to me from a workflow point of view etc, so it seems quite logical the way it currently works.
With regards to your 3rd question
The end points or caps not looking correct on custom brushes is to be expected sometimes. The round cap option on a regular plain line is just adding an additional bit of vector shape at the end of the line so if you had a custom brush it just wouldn't know what to extend to.