Graphic tablets allow you to use different expressions for more control over the brush stroke: all of them support pen pressure, some allow to use pen tilt, pen rotation, etc. The preview you see on your first image shows how the brush stroke would look like with the expressions enabled: it's set to Pen Pressure for opacity, flow and diameter. There's also a ...
Maybe it's stupid but I was looking like crazy to find a way on how to have shown default swatches on the fill panel and it's been really easy all the time. I clicked on the 4th icon at the bottom left on the fill panel (show swatch kids menu) and clicked on show all swatches. They just popped out.
You can try removing the brush stroke from the paths.
Merely click the Uniform item in the Brush Panel (Which really means "no brush stroke".)
Then click the brush you want to use.
This should apply the new brush with the default settings for that brush.
The short answer is, no, there's no way to "slide" or "offset" the pattern starting point on the path.
One possible workaround would be something like this:
Cut the path with the knife tool
Delete extra point, this moves the starting point to the spot where you cut the path.
Join the points and now you have a closed path again, with ...