Is my opponent allowed to use their robot to sit inside of my goal zone (the one without the protected area) and prevent me from placing cubes inside of the goal zone?
No, SG3 prevents them from contacting your goal zone. However, they could make a robot that surrounds the zone to block you from scoring.
Does the small goal have the same protection?
No, the smaller of the two goal zones lacks a protected zone (If you look at the field there are no tape lines). However, per < SG3 >, they still can’t mess with cubes that are actually in the goal zone, so as long as you can get them in there , they are protected without an actual protection zone.