Curiosity rover writes Morse Code of JPL everywhere it goes

They have very detailed physics-based models of terrain-wheel interaction ("terramechanics" is the general term). The tread depth must be one of the design parameters.

Slip is something they watch closely, and 100% slip is something they worry about a lot. I worked on a project using a small prototype in that same yard (metal wheels with metal tread) and was really f-ing hard to get the thing to slip with anything like a reasonable load and slope.

What happens if the ground moves?

For that matter why not use the same technique a laser/led mouse uses to know how far/fast it's been moved?

They could have one on each wheel and then another set of sensors pointing down, so wheel slippage vs ground movement at that axle.

As I understood it that's exactly what the camera is doing, and the morse code is just a helpful pattern with a known distance to make more precise measurements of ground speed.

With no patterns one could think of a situation where the surface sand/terrain would be displaced but your actual position was exactly the same.

Reasons I can think of include: the ground does not consist of a regular pattern of known length so image processing might be more difficult; looking out over the track allows more data points to be interpreted; an outward looking camera can fulfill more additional roles or be repurposed to compensate for other failed instruments more readily; a downward facing camera is more likely to gather images within the rover's own shadow; other outward facing cameras could use the same algorithm should the distance measuring camera fail.

Light heartedly off topic, but at a decelerators conference last year I had a chance to talk to one of the JPL Entry Descent and Landing peeps, and he mentioned that in Q&As, no single audience utterance carried more foreboding than 'Why don't you just...'

To answer your question more helpfully, having closed loop machine vision stuff is quite computationally intensive, though there is already some on the rover (to help negotiating obstacles or stop if it sees something like a trench (MERs would famously stop at the same time every day, thinking that their now-infront shadow was a pit)) but it's power intensive and expensive (both mass and cost of a dedicated odometry camera, for example) and the lighting conditions are very variable. The MERs developed a technique whereby they would 'shimmy' their steering every so many meters, and then with the rear facing camera snap a photo of their tracks (with shimmies) and infer distance travelled and slippage and so on. This stuff is often easier with humans in the loop as the terrain still throws up unknown unknowns when driving. Here's a pic of the tracks:

http://photojournal.jpl.nasa.gov/jpeg/PIA14129.jpg

I guess another point is that while it might be a perfectly sensible suggestion technically, there's only so much 'new' they want to risk per mission (the Entry Descent and Landing phase contained a whole lot of 'new' as we know) and they maybe just wanted to carry on doing it in a way they know works ok from before.