If you want the vertical cut of the line to have a certain height
This is very easy to do with skewed rectangles.
Move the two magenta rectangles in the left side so they snap to the inside of the yellow rectangle.
Change their color to the wanted green.
In the upper green rectangle, select the two anchor points in the right side with Direct Selection Tool.
What you need is not possible with Illustrator's path logic.
Paths need to in be a continuous, one-after-another anchor point sequence.
Instead, you can either group these, or convert to outlines, in which case you can join multiple shapes into a single shape, but you're losing the ability to edit these as paths (unless you keep an un-converted copy).
No branching path is possible. Use grouping.
Or outline and pathfinder for a outlined shape.
But this question opens a good learning opporunity:
The purpose of a vector file is not to make it possible to model whatever you want how you want it. Instead the purpose is to make a minimal set of features that are needed to render a particular feature to ...
You can see the path if you do Edit > Preferences > Tools > Node, and check the "Always show outline" option.
You can't easily have a path with an invisible stroke on one segment. A path only has one stroke attribute. I mean it would be possible with tricks like clipping paths or masks but you probably don't want to go down that route just ...
Let's assume your description of your situation is valid. Then your shape should be a single path (=a Bezier curve) which is filled with orange and has a black stroke. The black edge shouldn't be a rendered effect, it should be seen in the stroke and fill panel like in the next image (the nodes are visible because the node editing tool is active)
In the ...
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 ...