I need a genius...? 2D layer changes position when toggling 3D. How do I keep it where it is?

Place this Expression into the 2D layer's transform>position property. The name of the 3D layer this Expression is referencing should off course be changed to reflect the 3D layer you are intending to follow, in your comp. 

This only gives a representative 2D location of the 3D layer, without scaling/Z-position effects but it works well enough under certain use cases.

HTH

thisComp.layer("ImA3DLayer").toComp(anchorPoint)

If you want to manipulate a single flat wall, a planar track with mocha is much easier and more accurate than a 3D camera track. See this tutorial for a clean plate workflow with Photoshop:

I may be missing something but, if you used the 3D Camera Tracker:

• I would have made a flat Photoshop layer for the wall decorations.
• With the 3D Camera Tracker effect active, you can select multiple track points on the same plane on the wall. This will give you a target icon
• Right–click on the target and choose create solid. That solid will match teh plane of the walls surface and should stay in perspective as the camera moves through the scene.
• change the dimensions of the solid to match photoshop layer.
• Precomp the solid layer and "Leave all attributes"
• Open the Precomp and add the transparent layer from PS into it. 

That should give you a graphic that matches the perspective of the wall as teh camera moves through the scene.

As long as it's a 2D layer, it has no 3D coordinates and there's really not much more to say than that. AE simply has no way of retaining the position since it never knows where exactly the content of the image would be in terms of depth. It only assumes default zero coordinates. This has nothing to do with it being mathematically impossible or any of that, it's just how AE works. Of course it would be possible to craft convoluted expressions with a ton of toWorld(), toComp(), fromWorld() and so on to calculate something, but you'd still need a point of reference and at the very least this would mean placing several Nulls in the scene to calculate a plane or intersection. Otherwise you may want to refer to the various "projection mapping" tutorials out there. You have to align something first or else you're flying blind. In the end it's really just a matter of preference at which point your texture gets unwarped and re-applied, but eventually that is part of every workflow regardless of what tools are used.

Mylenium

Your logic is flawed and the workflow wrong. Exporting the PNG has already perspective baked in and it would be way too much work to fix any of that. the proper way to proceed is to unwarp the layer conventionally so that it matches whatever solid or pre-composition you have placed in your scene and then simply replace that. It's called reverse projection mapping. Since AE does not provide a way of direct texture/ projection baking, you have to kind of hack it by creating a solid that matches the area you want to cover, render this out as a reference matte and also export a still frame of your background. Then you use both in Photoshop to unwarp your image to a regualr rectangular shape based on teh reference points your matte provides. The same could be done in AE using tools like Bezier Warp, Corner Pin, Mesh Warp, CC Power Pin or advanced third-party plug-ins. Point in case: You realyl need to remove the perspective from the texture or else this becomes entirely unpredicatble and you'll work your fingers off trying to fix it. Only then can it be warped back using your camera tracker or corner pin 2D track.

Mylenium