How/why the hell....

[Sundance]

Why is it that, as AI, I can sometimes try to track someone and they'll go somewhere my cameras can't see and then immediately back into camera range and yet my camera will still be sitting where it stopped leaving me staring at space like a chump while they run off, without any 'ceased tracking' message?

The trick with this is to having the tracking window open, and when said player disappears and you somehow reestablish visual, that you can simply just click on the tracking window, press a letter on the keyboard and it will pop up, i.e: A for andy anderson. This happens with most listed action things. There is no cease tracking however.

On a semi-related note, what the hell am I supposed to do when someone asks me to open a door and I go track them but the nearest camera can't actually see them?

There is an action to open a door when the player is adjacent to that door, it is a fairly new addition. It's tied in with the textbox, but I haven't played AI myself to know it's effectiveness.
Failing that, you can always use the AI shell to come to the aid of a blindspot area (This is including in space, AI shells can use pods!)