Personally, either I edit the part if it's on a part present in the assembly, or I create a part (insertion) in the assembly and I work in the context.
No, it is not possible. The removal of material can be shared or propagated to the rooms. But for extrusions, you have to go to the assembly tab and deploy, insert a new component and then a new part and there you can make an extrusion or revolution that will be recorded in the assembly... But at any time you will be able to save the part outside of the assembly by opening it and making it saved as.
I always make the comparison with reality: when you make an assembly, you can possibly do a machining on it (removal of material like simultaneous drilling of 2 parts) but you can't add more.