I wonder if it got caught on a mountain or something equally robust it could stop the moon's orbit. I wonder what, aside from the tides, this would affect.
Are you suggesting that we reel in the moon to geosynchronous orbit? Ignoring our obviously insufficient mastery of the art of ropemaking etc, that's a quite entertaining thought experiment. Tidal forces while moving the moon closer would knead our whole geology into utter chaos once, then later tectonics could stabilise much more than possible with the moon in its current orbit. The weather would also be quite funny, with those daily, hours-long eclipses hitting the same regions over and over again, causing dramatic gradients in energy influx.
We are already doing that (or rather, the moon is doing it to us), more by changing earth's rotation than by changing distance. We just need to wait a couple billion years for earth to be tidally locked to the moon (and not just the other way around like it is now).
You would need quite the rope ladder in order to stop the moon's orbit.
Also, once you've stopped it, it would probably fall towards the Earth, because it is not at the right altitude to stay in geostationary orbit. That would be extremely bad.