Can everything be made cardboard-compatible? I had a lot of text here but I wrote the tl;dr and realized it said everything the paragraphs did: it's obviously possible to use cardboard for whatever, with relatively minor modifications, so, is there a custom rom or something that allows exactly this? I've dug around a bit and haven't found one. 

Also: if you're wondering what this would possibly be useful for, just imagine playing Canabalt or actually basically any other game with VR, without praying that the game devs get on the VR hype train. Immersion is fun. 

Also also: I know there are already somewhat similar questions on here, but I think that the answers given previously didn't really answer the question I have here.
No. There's a lot more to making a VR game than just displaying existing 3D content in a different mode, and not all existing content is even 3D. Your Canabalt example is an obvious case that won't work for the simple reason it isn't a 3D game. There's a lot of extra work needed to compute the extra view for a second eye, and that's a source-code change, not something that can be done automatically.

Even the 3D first-person games that would lend themselves to VR conversion aren't that simple. PC VR fans have tried patching graphics drivers to enable this with mixed results. The problem is, many games use some parts that aren't a simple 3D view: a 2D HUD, or drawing the first-person weapon in a different pass with depth disabled. These features all cause problems, and require game-specific fixes. On top of that, even when you make your target game "just work" in VR, it'll probably make you throw up. Interaction and motion in VR need extra thought to make immersion work without causing motion sickness.