<Would a water dump create an echo traveling at 5 meters per second (about 11 mph) relative to the Orbiter? For that matter, would an impact knocking loose a piece of tile?

For the first question, yes, an 11 MPH differential isn't all that high. It depends on how fast the water was going when ejected from the Shuttle. Kim would know this better than either of us, but it isn't hard for me to believe they'd eject the water somewhat forcefully to make sure it gets well away from the orbiter.

As for the second question, I'd guess that a debris impact significant enough to knock off a piece from the Shuttle would have pieces going faster than 11 MPH relative to the orbiter. If anything, that number seems on the low side.

The key thing about a debris impact is the relative velocity between the two parties. If the Shuttle and the debris were nearly in the same orbit, the relative velocity could be very small. However, an increase in the difference in inclination and orbital planes between the Shuttle and the debris leads to a higher relative velocity. Columbia was in a 39 degree inclination orbit. Not many other things are in an orbit with that inclination, so the relative velocity could be pretty high.

If there is a chunk up there, it should still be there. And they would have a pretty good idea exactly where. Are the radar guys looking? If it was a chunk of water ice, would it have ablated to nothing by now in the vacuum? If so, finding an echo a week or so later would make it almost certain to be a structural item.

Not necessarily. Some objects decay much more rapidly than others. It has to do with the relationship between the drag up in space and the mass of the object. An object with a low mass, say a piece of tile, would decay pretty rapidly. The direction of the object as it separated from the Shuttle could also be a factor. If it was going opposite the line of flight (slower), then it would drop into an orbit with a lower perigee, increasing the drag. If it was going faster than the Shuttle, it would settle into an orbit with a higher apogee, slowing decay. If it was more or less perpendicular to the Shuttle's line of flight, then there would be little if any change in drag.

Over the years, astronauts have dropped things from time to time. I remember seeing an entry in the space catalog for a screwdriver dropped during a spacewalk. Another time, there was a glove that somehow got loose. Some of these things can stick around for a long time. On the other hand, cosmonauts used to put their garbage into bags and jettison it. Those things tended to burn up quickly.