Rad fan - Check the fuse and the relay(both under the bonnet). Check with a 12v supply that the fan spins up.
You can bypass the relay by removing it, and bridging two connectors on the socket. Fan should run.
It's the brown sensor that sends the signal to the ecu and thence to the fan.
I can't really help with the auto box, other than my previous comments of new fluid and get it hot.
If the tickover is OK cold, but high on warmed up it does sound like the IACV may be stuck.