To save it as an .f3d, right clink on each component with a linked glif on it and "unlink". That is is. None of the other things you said is relevant to saving as .f3d
Local contexts are referring to references you made in your current design while editing the linked component "in place" (using the pencil glif to edit the linked component). They get saved in your linked component so if you open it separately you can still use the references for modeling.
when you use save from the file menu, everything gets saved to the cloud. nothing in your model is "on your computer". might have originated there, but it all get saved up when you named the model and saved. you can't have a live link to a file on your hard drive.
.f3z works just fine as a means of sharing a complete design with all the external references. I just don't like to open them from people I don't know b/c they can pack all sorts of weirdness into them, and I can't tell till after the upload is started.