I think the issue mostly stems from how you've assembled things here. In the below screenshot, I've isolated the design I'm assuming you're performing a Save Copy As. I've also Activated this component. Activating the component will filter the timeline down to the features that exist only in that component's timeline. These are the features that are 100% going to be carried with the component if you perform a Save Copy As.

The one thing that jumps out to me when I activate the component is that there are no joints, rigid groups or positional features in the component's timeline. It looks mostly like copy and pasting components.
When you're editing components, it's a best practice to first activate this component. This puts the features that make up that component in that component's timeline and not in the root component's timeline. This is comparable to editing a part in-place in traditional CAD softwares like Inventor or SolidWorks.
This makes a big difference when performing a Save Copy As since the timeline has a better chance of being preserved intact. I note that if I add an As-Built (rigid) joint and a Rigid Group to that component while it is activated, I can then perform the Save Copy As and Insert without any components changing position. See this design as an example.
http://a360.co/2z6HC1w
I would highly recommend checking out the Master Assembly Class for Fusion 360 in the below link (if you haven't done so already). It contains some really great tips about creating assemblies in Fusion.
http://au.autodesk.com/au-online/classes-on-demand/class-catalog/2016/fusion-360/pd19577#chapter=0
I hope this information helps. Please let me know if you have any quesitons.
Thanks,
Nathan Chandler
Principal Specialist