[Physicist]

But I find it hard to believe that any other intelligences exist since some evidence of another civilization hasn't yet floated by, been received via radio wave, or even been carved on a rock somewhere yet.

But when you consider the percentage of the galaxy we've explored, whether it be examining rocks or radio waves, we've explored almost none of it. If there were another civilization with a radio luminosity just like Earth's, how close do you think it would have to be for us to see it? If there were a ten thousand alien soda pop bottles on the moon, how many Apollo missions would it take to find one?

[asgardshill]

But when you consider the percentage of the galaxy we've explored, whether it be examining rocks or radio waves, we've explored almost none of it.

True. But as I noted earlier, intelligence is curious. We would have been contacted already (or at least scrutinized with the alien equivalent of a kid with a big magnifying glass on a sunny day).

[Physicist]

To answer my own questions:

If there were another civilization with a radio luminosity just like Earth's, how close do you think it would have to be for us to see it?

Far less than one light year, if we didn't know where to look. If we did know where to look, perhaps a light year.

If there were a ten thousand alien soda pop bottles on the moon, how many Apollo missions would it take to find one?

Assuming a random distribution of bottles, and assuming that each mission could exhaustively search 10,000 square meters, it would take roughly 200,000 missions to have a 50% chance of finding one.