If you removed the earth, except for the moon the solar system would change very little.
Sure if the earth came between Jupiter and it's moons there might be some chaos, but the likelyhood of that is minimal. Even if it did happen we would notice, but the earth massed black hole would be very unlikely to make it into the inner solar system at that point.
The earth is a minor gravitational force in the solar system, most likely the effects would take significant time to notice... well after the nearest approach. Small gravitational tugs take many samples and significant time to notice. For example the outer solar system still doesn't add up... and we have no idea why.
The outer solar system is pretty well accounted for. For many years we thought it wasn't, and kept hunting for another (sufficiently-massive so not Pluto) planet beyond Neptune. The discrepancy turned out to be an overestimation of Neptune's mass, which was resolved by Voyager and subsequent observations. We've now ruled out anything of Neptune's mass or greater existing out to at least 100 AU and likely several multiples of that.
Probably not. It would be something that would likely have to be observed over multiple complete orbital cycles and basically somebody basically run into it by accident. By which, of course, the earth in this hypothetical, would be long dead.
