So am I correct in thinking that essentially the intended emphasis of גם is bound to whatever is immediately after it?
For example, would it be the case that one could say:
אני אופה גם עוגות
To mean, "[In addition to the other things I bake,] I also bake cakes?"
Or would it instead be that "אני גם אופה עוגות" would require context to differentiate the above meaning from "I also bake cakes" (as opposed perhaps to merely eating them)?
I'm asking for clarification on the last bit specifically because I thought I understood until I was presented with this exercise. :)
Your initial conjecture is true - גם tends to emphasize what's immediately after it - but it's a weak tendency, I think, intonation and context can quite easily override it.
The exercise you link to is quite artificial, in both languages I think, so I wouldn't try to learn such subtleties from it...