In a given CATIA assembly, I have my master geometry in a part.
1) When publishing sketches, lines, and edges for other parts to use, should I publish into the master geometry part level or publish to the assembly level?
I publish at the part level so those published elements can be used in constraint creation if the part is put in other products as well. For example, for a fastener part I would publish the axis at the part level since that fastener will be constrained in multiple assemblies.
The same as Drew said.
What do do you mean by assembly level? At assembly level I activate a part and then extract geometry that I need from another part.