Maybe the falloff in output has convinced them our civilization is dying and not worth contacting.

That seems unlikely to me. Why wouldn't at least a few alien civilisation keep broadcasting radio signals just to let others know they are there too ?

Simple, you just keep tapping it with a rubber hammer until you get alignment.

I don't know if you need a "trick"... Just several orders of magnitude more precision?

Actually, I think this is less of a problem than you think. Lasers have significant divergence, see for example the Marcy paper an astro-ph that was referenced elsewhere in this thread. Telescopes can already track targets at the diffraction limit, so it's no different to send a laser.

What is different is that you have to know the relative proper motion of the target, since you have to "lead" the target just like you do when skeet shooting. I don't know whether the lead angle is significant, but proper motions of nearby stars have been measured (e.g. by Hipparcos http://en.wikipedia.org/wiki/Hipparcos) and calculating the resulting lead angle isn't difficult.