Since Alfred and Batcave Computer count as characters, what happens when you try to Capture say... Alfred? Does it remove the whole Batcave from the game? And then what happens with the rescue if you end up moving away to the middle of the map...will this be a way to "move" the Batcave away from the starting area?

I believe they cannot be captured since they cannot be moved or placed. It is similar to how they cannot become pilots.

How does capture ability have anything to do with being moved or placed?

When a character is captured, that character is removed from the map. When the captive is released or rescued it is put back on the map in the area where it was released or rescued. While the Capture rules never uses the word place like vehicle rules do, the two seem similar enough that I believe they would both work the same way.