How can I composite these 3 Layers? I'm attempting to add a couple of filters to a render layer that sits between two others. I suspect my problem arises from a misunderstanding of how layers work, as my "flames" layer renders on the "door" layer as well as its own... Anyway I can't get a result at all. If anyone can give me some pointers, I'd be much obliged.
You can't compose this way because you use Glass shadier on the window. Glass isn't just transparent color - it have refraction and reflection, that's why it renders non-transparent. So you can try different things:

1. Just remove window or move it on different layer.

Then use Mask layer. If I move window on this layer:

 

The setup for render layer flames will look like (pay attention to the mask layer option):



and door of course:



result will be look like:



2. add glow effect by threshold 

in this case you should not to separate render layers. Render all in one frame and in glare node set threshold to value between 0 and 1