Could the Earth use gravitational lensing / bending of light to see it's own bottom? Specifically, I have often wondered whether gravitational lensing might be able to (theoretically) be used in order to see light reflecting off the Earth that has zoomed off into space and been bent by one object after another until it comes full circle back to Earth again. Given the distance the light would have travelled, could we not use this technique to peer back through time at our own planet?


Is it reasonable that this could happen given the size of the universe and the mass and distribution of objects within it?
Given the restraints above, would be be able to estimate how far back in time we are looking?
Given the above, how bright would something have to be in order for its light to travel this distance and still be detectable?
Whimsical question. Similarly, you could reason that someone on a planet far, far away would be holding up a mirror, such that we could see our own reflected past again.

Unfortunately, both options are practically infeasible, even if the light would be undistorted and redirected perfectly towards us. Remember that (1) Earth is small, (2) the reflected signal is weak as it is, (3) any object capable of reflecting this back towards us would be a black hole, the closest of which is further away than most known exoplanets, and the signal falls of with $r^{2}$ with $r$ twice (!) the distance to the object doing this lensing.

Now, if we somehow were able to observe it anyway, it would be straightforward enough to estimate the look back time - the distance to the lens would presumably be fairly well known. 

How bright it would have to be is not really a sensible question - there are too many variables in play for that (which (hypothetical future) telescope are we using, what are the observing conditions, how far away is the source, what kind of source are we talking about...). And at the end of the day, the brightness of your source is not even the biggest problem; finding a suitable lens is.