You may be on to something here. The Giant impact hypothesis remelted a good part of Earth and splashed off the Moon. Instead of a bunch of little watery asteroids, bash it with one big enough to liquify the parts that have cooled and seized up.
Yes, this requires really long-term planning.
You've just described the flaw in the plan: who would buy such a thing? Boeing isn't going to build them out of the goodness of their heart. The people who put satellites up would benefit from clearing out the orbit they want to use, but that is nowhere near as simple as putting one of these sweepers into that orbit, only retrograde, because of all the different orbits of the debris that intersects it. It would be ridiculously expensive to try to clear one orbit for one satellite.
What we have here is an example of a tragedy of the commons because no one owns the valuable space where satellites live. Perhaps an international consortium of space-faring nations could claim it all and put orbits up for auction, then use the proceeds to clean it up?
If the ship you are searching for has been in one spot for more than three minutes when you jump in, then it can be seen immediately. More precisely, for whatever time t it has been sitting, it can be seen at distance d = ct, where c is the speed of light, because light from it has filled a sphere that large. Meanwhile the ship that jumped in starts emitting infrared and reflecting starlight to create the lightsphere in which it is visible.
So the cylons could jump to three light minutes out, shoot beam weapons for 2:59 and then jump away before the Colonials even know they've arrived.
So here's another thing: if the dradis works ftl, then there ought to be a way to use that same technology to create a tight, high-powered beam of "dradis radiation" that works at the same speed as dradis. Then the same equation applies but in which c is the speed of dradis.