1) There is not currently a way to make that icon disappear, no.
2) When a computer goes to sleep it stops running any programs, so no, this is not possible. Your monitor can go to sleep and your hard drives can spin down, but the computer going to sleep is basically the same as turning it off when it comes to any of the programs running.
3) The green iSight light is something that is controlled by the hardware as far as we know. When the camera becomes active, the light turns on.
4) Some sort of recording on motion has been requested in the past and is currently on our list of potential features to implement in the future.
Thanks!