What sensor can pick up really tiny movement without touching the thing?

I've chased this exact kind of thing before and the trick is to stop trying to sense tiny movement directly and amplify it mechanically instead. That's how you get to a hundredth of a millimetre with pocket money parts.

What I'd do is stick a small mirror tab next to the crack, aim a cheap laser pointer at it, and let the reflected dot land on the opposite wall. When the wall moves even a hair, the tab tilts slightly and the dot slides a long way across the room. The further the throw, the bigger the movement you see, so the longer your hallway the better. Then point an ESP32-CAM at the far wall and log the dot position. A camera doing subpixel tracking on a bright dot against a plain wall is well within what that little board can handle, and it's the same idea as your laptop plan but it sips power and can run off a power bank for months. I've used this setup to check wether a gatepost was shifting and it worked a treat.

Another cheap route is a lever. Mount a thin stick on a screw pivot so one end sits against the crack and the other end sticks out. If the free end is ten times further from the pivot than the end touching the wall, movement gets magnified ten times, and a cheap dial indicator or even the camera idea can read the free end. Dial indicators with 0.01mm markings cost next to nothing, though reading them into an Arduino needs a photo or a hacky mod.

One thing to expect: the room's temperture will move your readings more than the crack does. Everything in a house breathes with day and night, so log the time and roughly what the room temp was doing and look at the trend over weeks, not individual readings. And honestly, a few months is the right window anyway, since cracks often cycle with the seasons rather than creeping steadily.

I find the whole insistence on logging room temperature slightly overboard. While ambient shifts certainly happen, the idea that thermal expansion across an entire structural element is going to interfere more with a 0.01 mm shift than the actual movement you are looking for seems unlikely. If a building were expanding enough due to temperature to cause a measurable movement on that scale, it whould likely be an issue of structural integrity long before you could see a small crack developing. Focus on the mechanism itself instead of the indoor air flow.

If you relly want to nail this without relying on a camera's ability to track a pinprick, you could look at a linear displacement sensor. The key is making the sensor itself sensitive to the tiny shift, rather than trying to move a dot across a wall. You mount a small probe across the crack, and the sensor measures the physical change in distance directly as it happens. It's simple contact measurement that bypasses the optical tracking whole discussion. I found those types of transducers work well for monitoring foundation shifts. The main thing to check is the sensitivity rating; you'll want one that registers a fraction of a millimeter or less. Just make sure the mounting bracket keeps the sensor perfectly perpendicular to the wall. Any wobble in the setup will swamp the tiny movement you are actually trying to log.

I used a similar setup to monitor a bridge joint shift and realized that reading a cheap dial indicator is a nightmare due to parallax. I ended up using a very small optical encoder attached to the lever arm's pivot point. That one tiny component made all the difference for me because it gave me digital, repeatable readings for the minute rotations, which I coud easily translate into movement distance.

If you go the lever route, the question is how you capture that tiny rotation at the free end. Once you have that encoder reading, you're looking at direct, digitized movement that’s free from the need for any visual processing.

I'd bet on looking into piezo discs first. If what's happening is mostly minor settling or even just thermal stress releasing in small jolt, those things are great at catching tiny, sudden shifts. I used them for a project monitoring a foundation shifting a bit, and they don't need the same setup as a full displacement sensor. They just need to translate that tiny mechanical change into a usable signal for your ESP32.