Is it possible to get to the E3 area (aka the wall, aka the first level) in the Explore mode? I went through the Breach, but it seems that getting to the Cosmodrome wall is impossible because there's a gate in the way.

Yeah, I think you can only do some exploring in the intro level. I wouldn't worry about the Fallen that your Ghost keeps worrying about. I was just minding my own business for a few minutes and heard no roars or gunshots directed at me.