Your layers are intersecting because they are intersecting in 3D space. Look at a left or right view and you'll see the layers cutting into each other. If you want them not to intersect then you have to move them. The most common thing to do is to change the Z distance. It sounds like you don't understand 3D space. AE works like all 3D apps except that the layers are limited to 2D planes. If they intersect in the 3D world then they intersect. If one moves behind another then it is behind. The layer order does not make any difference unless you start inserting 2D layers or adjustment layers between 3D layers.
I have no idea what you mean when you say "I was going to insert an image - but when I click the camera icon - all I get is a big grey box that says 'Insert Image' - . . ." because you don't insert images into After Effects, you import them to the Project panel and then you drag or add them to the timeline. There is no camera icon in AE that has an Import function. What exactly are you doing and what app are you using to Insert something by clicking on a camera icon. There are no Insert functions in After Effects.
I suggest you look for some basic AE tutorials on 3D space. There are dozens if not more that give you the basics.
Sorry - the camera icon reference was to this forum. When I click the camera icon to insert an image - all I see is a grey box. Nothing to do with AE.