I would imagine the water channel would re-freeze fairly rapidly, so you'd end up with a "bubble" of liquid water around the thing slowly melting itself down.
The ice clearly moves, as Europa's surface isn't just same as an airless, solid ball-of-rock's default: craters, but has all sorts of features that reshape the surface, so given enough time, it'll equilibrate. (What 'enough' means is left as an exercise to the reader). There are papers[1] that discuss the ice properties, but it's hard to get a specific answer out of them. There have to be tons of research papers out there about the design criteria for melt-drill probes like this, for Europa, Enceladus, and others.
