OK???? you can use sketches from any component in another sketch, in the tree you just have that components sketch turned on and the component for the new component turned on you just do a plane and project the sketch onto that plane what is inside the new component what will give you a new sketch what is a projection from another sketch once you have project what lines or what ever are needed you just hit OK, turn the sketch used for the projection from off.
you can import other sketches straight into a component copy and past if you have a component turned on it's the only place it can go that's why you have it turned on so you know where to stick it.
if you save copy as, and there is only a body in the component of course it will just have a body in it, if a component has a sketch, plane and body in it and you save copy as it save everything inside that component. just tested work as it should.
what you are better of doing is copy and past that will save everything inside the component and keep the component intact save copy as does not do this yet.
in that new file you can actually drag the body into the empty component but the sketch is still outside of the component in the Browser.
yes then you redefine that sketch into the new component by selecting the face of the body it came from as the plane.
If you redefine a sketch plane for a body that was created before a component(blank or otherwise) was created, during that operation the other components in the browser will disappear so you can't select their geometry.
correct. then you move the sketch or body in the time line, or you can move the time line to before that body was done making a new component then move the time line back to the end and move it.
all off this I can show you very easily I do it everyday, have a look at this vid for how to copy and past in to a new Design and how to fix it. you can do the same inside a Design
https://www.youtube.com/watch?v=XLN4AV16MQ8
if you wont I can just do a vid on save copy as.
also some of thing you don't know how to do are in the fusion 360 youtube channel.
I learn all this from the top 5 fusion user's and the ADSK fusion employees